Die Grundlehren der
mathematischen Wissenschaften in Einzeldarstellungen
Band 181
J. L. Lions • E.Magenes
Non-Homogeneous
Boundary Value Problems
and Applications

Die Grundlehren der
mathematischen Wissenschaften
in Einzeldarstellungen
mit besonderer Berucksichtigung
der Anwendungsgebiete
Band 181
Herausgegeben von
J. L. Doob • A. Grothendieck . E. Heinz • F. Hirzebruch
E. Hopf • W. Maak . S. MacLane . W. Magnus -J. K. Moser
M. M. Postnikov . F. K. Schmidt.. D. S. Scott. K. Stein
Geschdftsfuhrende Herausgeber
B. Eckmann und B. L. van der Waerden

J. L. Lions • E. Magenes
Non-Homogeneous
Boundary Value Problems
and Applications
Translated from the French by
P. Kenneth
Volume I
Springer-Verlag Berlin Heidelberg New York 1972

J. L. Lions E. Magenes
University of Paris University of Pavia
Title of the French Original Edition:
Problemes aux limites non homogenes et applications (tome I)
Publisher: S. A. Dunod, Paris 1968
Translator:
P. Kenneth
Paris
Geschaftsfuhrende Herausgeber:
B. Eckmann
Eidgenossische Technische Hochschule Zurich
B. L. van der Waerden
Mathematisches Institut der Universitat Zurich
AMS Subject Classifications (1970)
Primary 35J20, 35J25, 35J30, 35J35, 35J40, 35K20, 35K35, 35L20
Secondary 46E35
ISBN-13:978-3-642-65163-2 e-ISBN-13:978-3-642-65161-8
DOI: 10.1007/978-3-642-65161-8
This work is subject to copyright. All rights are reserved, whether the whole or part of the material
is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction
by photocopying machine or similar means, and storage in data banks. Under § 54 of the German
Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount
of the fee to be determined by agreement with the publisher.
© by Springer-Verlag, Berlin • Heidelberg 19/2. Softcover reprint of the hardcover 1st edition 19/2
Library of Congress Catalog Card Number 71-151407

Preface
1. We describe, at first in a very formal manner, our essential aim.
Let 6 be an open subset of RM, with boundary d(9 .In 6 and on d(9 we
introduce, respectively, linear differential operators
P and QJt O^j^v.
By "non-homogeneous boundary value problem" we mean a problem
of the following type: let / and gJ} 0 ^ / ^ v, be given in function
spaces F and GJt F being a space "on 0" and the G/s spaces "on 30";
we seek u in a function space ^ "on 0" satisfying
(1) P« = / in (9,
(2) ft« = gj on 30, 0 £ / £ v((1)).
Qj may be identically zero on part of d&, so that the number of
boundary conditions may depend on the part of d(9 considered2.
We take as "working hypothesis" that, for feF and gjeGj,
the problem (1), (2) admits a unique solution ue<%, which depends
continuously on the data3.
But for all linear problems, there is a large number of choices for the
spaces tft and {F; Gj} (naturally linked together).
Generally speaking, our aim is to determine families of spaces tft and
{F\ Gj}, associated in a "natural" way with problem (1), (2) and
convenient for applications, and also all possible choices for <% and {F\ Gj}
in these families.
Let us make this explicit by means of two examples, chosen as the
simplest possible ones, but which already demonstrate the utility of
non-homogeneous problems.
((1)) The Qj's will be called "boundary operators". Such problems are called
non-homogeneous because if we consider the setting of unbounded operators, then
Pu = f, u€.D(P) (= domain of P) implies null boundary conditions; hence
a certain difference between / and the boundary data gj.
2 This will obviously be the case for most problems of evolution.
3 At least in general; for elliptic problems uniqueness conditions will not be
satisfied, but in any case we shall deal with operators with indices and therefore
still have uniqueness on passing to the quotient by finite-dimensional subspaces.
We shall verify that this "working hypothesis" is satisfied in each particular
situation.

VI
Preface
2. Examples
2.1 For 0, we take an open subset Q of R", with boundary r, and
for P the operator A, A = Laplacian; we take v = 0 and Q0 = identity.
Then, the problem corresponding to (1), (2) is the classical Dirichlet
problem for A:
(3) Au = f in Q,
(4) u = g0 = g on r.
We then ask in what spaces f and g may be chosen so that (3), (4) admits
a unique solution (in an appropriate sense).
Classical answers are furnished by potential theory and the "Dirichlet
principle ": for example, we may choose /, g and certain of their derivatives
in Q and r respectively to be square integrable and obtain u, the solution
of (3), (4), as well as certain of its derivatives to be square integrable
in Q.
Therefore, a natural family for problem (3), (4) must be (if we limit
ourselves to "L2 theory", that is Hilbert theory) the Hilbert family
of the Sobolev spaces HS(Q) and Hs(r), where HS(Q) (resp. Hs(r))
is, if s is an integer ^ 0, the space of us such that u and its
derivatives (in the sense of distributions) up to order s are square integrable
in Q (resp. P) (this definition is generalized to all real s by introducing
the derivative of order s by Fourier transform; see Chapter 1).
Here is one of the results we shall prove (see Chapter 2):
Let s be any real, non-negative number] if f e Hs (Q) and g e Hs+3/2 (J1),
then there exists a unique u eHs+2(Q) which is a solution of (3), (4)
(having given an appropriate sense to (3) and (4) separately, by a natural
generalization of the classical definitions). More precisely: the operator
u -» {Aw, u\r} is an isomorphism of
HS+2(Q) onto HS(Q) x Hs^2(r).
It must be pointed out that the derivatives of non-integer order
necessarily enter the problem if we want the optimal result for each s,
since s + 2 and s + § cannot be integers simultaneously!
Furthermore, it is equally natural to study the case "negative s",
since many problems deal with (3), (4), where, for example, with / = 0,
g is very irregular: such as g square integrable on T (this is the case
for optimal control theory), or g = the Dirac-mass at the point x0eT
(then the solution u yields the Poisson kernel of the problem) or more
generally g = an arbitrary distribution on r.
It is still possible to solve problem (3), (4) when s is a negative real
number, the spaces °tt and G remaining of the same type (i.e. °tt = Hs+2 (Q),
C = Hs+312 (r), but with negative s) and the space F being an appropriate

Preface
VII
subspace Ss (Q) of Hs (Q) consisting of elements which do not grow too
rapidly "in the neighborhood of r" (see Chapter 2, Sections 6 and 7).
It follows that (3), (4) is solvable with g an arbitrary distribution
on r, since then g necessarily belongs to a space Hs(r), for an
appropriate s.
In fact, in volume 3 of this book, we shall see that g may belong to
the space of analytic functional on r (and this space is the most general
for which, at least for / = 0, it is possible to give meaning to problem
(3), (4)).
2.2 As a second example, we consider the heat operator
dt
in
= Qx]0,T[ c R"+i;
the part of the boundary d(9 on which boundary conditions are given
splits up into
D and Z = rx]0,T[.
Then, a problem corresponding to (1), (2) is
(5) ^L-AxU = f in &,
dt
(6) ' u(x,0) = u0(x) in Q,
(7) u = g on E.
One of our aims is to obtain the largest possible families of spaces
for /, u0 and g such that (5), (6), (7) admits a unique solution, in an
appropriate sense. We shall see, in Chapter 4 of Volume 2, that a
"natural" Hilbert family of spaces tft for the solution is the family H2StS{(9),
with s any real number, where H2S,S((9) is, if s is a non-negative
integer, the space of us such that u and its derivatives up to order s in t
and up to order 2 s in x are square integrable in 0. The family H2S,S(&)
plays, for problem (5), (6), (7), an analogous role to the family HS(Q)
for problem (3), (4). Also, remarks analogous to those we made for
problem (3), (4) are valid for problem (5), (6), (7).
3. We now specify which are the principal systems {P, Qj} studied
in this book.
3.1 We consider the case where P is an elliptic operator (denoted
by A) and the ()/s are normal boundary operators (denoted Bj), where
A and Bj verify suitable ellipticity conditions (see Chapter 2).

VIII Preface
3.2 We consider the case where
dt
a parabolic operator with suitable boundary conditions (Chapters 3
and 4).
3.3 We also consider the cases
P = -?L+A
dt2
and
*-hiA-
where A is a self-adjoint elliptic operator, and still with suitable boundary
conditions (Chapters 3 and 5).
4. For all these problems, we proceed systematically as follows (except
for possibly different techniques):
(i) we study the regularity of problem (1), (2), i.e.: assuming the
data / and gj to be regular (in a sense to be specified), we study the
corresponding regularity of u\
(ii) by transposition of (i) (for the "adjoint problem") we deduce
therefrom (with a suitable technique and in particular the obtainment
of "trace theorems") the solution of problem (1), (2) for data belonging
to spaces of distributions;
(iii) by interpolation between (i) and (ii), we obtain "intermediate"
results.
Of course, the systematic setting-up of such a program is an enormous
task and many possibilities had to be put aside (we have formulated
them in lists of problems in the last sections of each chapter).
In general, we consider for (i):
in volumes 1 and 2: data which are finitely often differentiable in
the sense of L2 (spaces such as HS(Q), H2S-S\@)} Hs(r), . . .)
in volume 3: analytic data or data belonging to suitable Gevrey classes.
5. As we have seen, the present volume depends on regularity
theorems in " differ entiable in the sense of L2" spaces (Sobolev spaces).
Therefore, the basic tools are:
— Sobolev spaces constructed on L2,
— the theory of interpolation of corresponding spaces.
This is the subject of Chapter 1, where we study interpolation only
for the Hilbert cases; the introduction of interpolation between (non-

Preface
IX
"hilbertizable") Banach spaces and its applications to Sobolev spaces
constructed on Lp, p 4= 2, would have complicated this work
considerably.
Once in possession of these tools we need to prove regularity
theorems (stage (i)) and then to implement stages (ii) and (iii).
This is done for the situations described in Section 3, above. Let us
be more precise.
The elliptic case is the subject of Chapter 2. Stage (i) is studied
completely by the method of J. Peetre [2], under the hypotheses that A
is properly elliptic and that the B/s cover A in the sense of Lopatinskii-
Shapiro and Agmon-Douglis-Nirenberg.
Stages (ii) and (iii) follow our previous papers on these subjects:
see Lions-Magenes [1], [2] and [3] (where we also study the Lp case,
for 1 < p < oc, which we disregard here for p + 2).
" Variational" evolution operators and their applications are studied
in Chapter 3.
Partial differential equations of evolution are studied in more detail
in Chapters 4 and 5 of Volume 2 and applications to optimal control
theory in Chapter 6 of Volume 2. Other applications will be given in
Volume 3.
6. We have made an effort to make the book readable in "local"
fashion; indications about the logical relations between the different
subjects are given at the beginning of each chapter.
7. Each chapter ends with a section of comments and a section of
problems.
The comments give bibliographical indications, which, although
numerous, by no means cover the subject. This is especially the case
for research work cited in the comments but not studied in this book.
The rather large number of problems to which we call attention
are very unequal in difficulty. For cases where results of type (i) are
already available, the execution of stages (ii) and (iii) may offer great
technical difficulties if one looks for optimal results, but is certainly
much more accessible if one is satisfied with results in the
"neighborhood" of the optimal results. Situations for which the results of
type (i) are lacking (and we indicate a number of such problems) may,
of course, be much more difficult.
We wish to thank C. Baiocchi, M. S. Baouendi and G. Geymonat
for reading various parts of the manuscript and giving us their
comments.
Paris/Pavia, July 1967
J. L. Lions E. Magenes

Preface to the English Translation
The present translation follows the French edition without change,
except for some corrections which were suggested to us by the remarks
of M. S. Baouendi, G. Geymonat, C. Goulaouic and P. Schapira, to all
of whom we express our sincerest thanks. We have added a
complementary bibliography. We also wish* to thank P. Kenneth for his excellent
work of translation.
Paris/Pa via, October 1971
J. L. Lions E. Magenes

Contents
Chapter 1
Hilbert Theory of Trace and Interpolation Spaces
1. Some Function Spaces 1
1.1 Sobolev Spaces 1
1.2 The Case of the Entire Space 4
1.3 The Half-Space Case 6
1.4 Orientation 8
2. Intermediate Derivatives Theorem 9
2.1 Intermediate Spaces 9
2.2 Density and Extension Theorems 10
2.3 Intermediate Derivatives Theorem 14
2.4 A Simple Example 18
2.5 Interpolation Inequality 19
3. Trace Theorem 19
3.1 Continuity Properties of the Elements of W(a,b) 19
3.2 Trace Theorem 21
4. Trace Spaces and Non-Integer Order Derivatives 23
4.1 Orientation. Definitions 23
4.2 "Intermediate Derivatives" and Trace Theorems 24
5. Interpolation Theorem 27
5.1 Main Theorem 27
5.2 Interpolation of a Family of Operators 27
6. Reiteration Properties and Duality of the Spaces [X, Y]q 28
6.1 Reiteration 28
6.2 Duality 29
7. The Spaces H*(RH) and H'(r) 30
7.1 Hs(Rn) -Spaces 30
7.2 Traces on the Boundary of a Half-Space 33
7.3 H*(r)-Spaces 34
8. Trace Theorem in Hm(Q) 38
8.1 Extension and Density Theorems 38
8.2 Trace Theorem 39
9. The Spaces H*(Q), Real s :> 0 40
9.1 Definition by Interpolation 40
9.2 Trace Theorem in HS(Q) 41

XII
Contents
9.3 Interpolation of Hs(£)-Spaces 43
9.4 Regularity Properties of Hs (^-Functions 45
10. Some Further Properties of the Spaces [X, Y]0 47
10.1 Domains of Semi-Groups 47
10.2 Application to Hs(Rn) 51
10.3 Application to Hs(0, oo) 54
11. Subspaces of HS(Q). The Spaces HS0(Q) 54
11.1 HS(£)-Spaces 54
11.2 A Property of HS(Q), 0^s<J 57
11.3 The Extension by 0 outside Q 60
11.4 Characterization of H^Q) -Spaces 62
11.5 Interpolation of H$(£)-Spaces 64
12. The Spaces H~s(Q)t s>0 70
12.1 Definition. First Properties 70
12.2 Interpolation between the Spaces H~S(Q), s>0 71
12.3 Interpolation between Hft(D and H-S*(Q), s, > 0 72
12.4 Interpolation between HS^(Q) and H~S*(Q), st > 0 73
12.5 Interpolation between HS*(Q) and (HS*(Q))' ,76
12.6 Interpolation between H*/(Q) and (//*(£))' 77
12.7 A Lemma 79
12.8 Differential Operators on H*{Q) 85
12.9 Invariance by Diffeomorphism of Hs (Q)-Spaces 85
13. Intersection Interpolation 86
13.1 A General Result 86
13.2 Example of Application (I) 87
13.3 Example of Application (II) 37
13.4 Interpolation of Quotient Spaces 90
14. Holomorphic Interpolation 91
14.1 General Result 91
14.2 Interpolation of Spaces of Continuous Functions with Hilbert
Range 94
14.3 A Result Pertaining to Interpolation of Subspaces 96
15. Another Intrinsic Definition of the Spaces [X, Y]0 98
16. Compactness Properties 99
17. Comments 103
18. Problems 106
Chapter 2
Elliptic Operators. Hilbert Theory
1. Elliptic Operators and Regular Boundary Value Problems 109
1.1 Elliptic Operators 109
1.2 Properly and Strongly Elliptic Operators 110
1.3 Regularity Hypotheses on the Open Set Q and the Coefficients of
the Operator A Ill
1.4 The Boundary Operators 112

Contents
XIII
2. Green's Formula and Adjoint Boundary Value Problems 114
2.1 The Adjoint of A in the Sense of Distributions or Formal Adjoint 114
2.2 The Theorem on Green's Formula 114
2.3 Proof of the Theorem 115
2.4 A Variant of Green's Formula 120
2.5 Formal Adjoint Problems with Respect to Green's Formula . . 121
3. The Regularity of Solutions of Elliptic Equations in the Interior of Q 121
3.1 Two Lemmas 121
3.2 A priori Estimates in R" 123
3.3 The Regularity in the Interior of Q and the Hypoellipticity of
Elliptic Operators ' 125
4. A priori Estimates in the Half-Space 127
4.1 A new Formulation of the Covering Condition 127
4.2 A Lemma on Ordinary Differential Equations 130
4.3 First Application: Proof of Theorem 2.2 133
4.4 A priori Estimates in the Half-Space for the Case of Constant
Coefficients 136
4.5 A priori Estimates in the Half-Space for the Case of Variable
Coefficients 142
5. A priori Estimates in the Open Set Q and the Existence of Solutions in
Hs(^)-Spaces, with Real s :> 2m 148
5.1 A priori Estimates in the Open Set Q 148
5.2 Existence of Solutions in Hs(£)-Spaces, with Integer s 7> 2m. . 152
5.3 Precise Statement of the Compatibility Conditions for Existence 155
5.4 Existence of Solutions in Hs(£)-Spaces, with Real s^ 2m ... 165
6. Application of Transposition: Existence of Solutions in Hs(£)-Spaces,
with Real s <; 0 166
6.1 The Transposition Method; Generalities 166
6.2 Choice of the Form L 167
6.3 The Spaces S(Q) and D*A(Q) 170
6.4 Density Theorem 173
6.5 Trace Theorem, and Green's Formula for the Space DSA(Q), s <J 0 175
6.6 Existence of Solutions in D^(£)-Spaces, with Real s :g 0. . . . 177
7. Application of Interpolation: Existence of Solutions in Hs (£)-Spaces, with
Real s, 0< s < 2m 180
7.1 New Properties of S^J-Spaces 180
7.2 Use of Interpolation; First Results 185
7.3 The Final Results 187
8. Complements and Generalizations 191
8.1 Continuity of Traces on Surfaces Neighbouring T 191
8.2 A Generalization; Application to Dirichlet's Problem 194
8.3 Remarks on the Hypotheses on A and Bj 195
8.4 The Realization of A in L2 (Q) 196
8.5 Some Remarks on the Index of ^ 198
8.6 Uniqueness and Surjectivity Theorems 199

XIV
Contents
9. Variational Theory of Boundary Value Problems 200
9.1 Variational Problems 200
9.2 The Problem 203
9.3 A Counter-Example 203
8.4 Variational Formulation and Green's Formula 204
9.5 "Concrete" Variational Problems 207
9.6 Coercive Forms and Problems 209
9.7 Regularity of Solutions 212
9.8 Generalizations (1) 212
9.9 Generalizations (11) 214
10. Comments 216
11. Problems 225
Chapter 3
Variational Evolution Equations
1. An Isomorphism Theorem 227
1.1 Notation 227
1.2 Isomorphism Theorem 230
1.3 The Adjoint A* 230
1.4 Proof of Theorem 1.1 230
2. Transposition 231
2.1 Generalities 231
2.2 Adjoint Isomorphism Theorem 232
2.3 Transposition 232
3. Interpolation 233
3.1 General Application 233
3.2 Characterization of Interpolation Spaces 233
3.3 The Case "0 = J" 234
4. Example: Abstract Parabolic Equations, Initial Condition Problem (1) 234
4.1 Notation 234
4.2 The Operator M 235
4.3 The Operator A 237
4.4 Application of the Isomorphism Theorems 238
4.5 Choice of L in (4.20) 239
4.6 Interpretation of the Problem 241
4.7 Examples 243
5. Example: Abstract Parabolic Equations, Initial Condition Problem (II) 255
5.1 Some Interpolation Results 255
5.2 Interpretation of the Spaces 0112, 0\J2 257
6. Example: Abstract Parabolic Equations, Periodic Solutions 258
6.1 Notation. The Operator A 258
6.2 Application of the Isomorphism Theorems 259
6.3 Choice of L 259
6.4 Interpretation of the Problem 260
6.5 The Isomorphism of &il2 onto its Dual 261

Contents XV
7. Elliptic Regularization 261
7.1 The Elliptic Problem 261
7.2 Passage to the Limit 262
8- Equations of the Second Order in / 265
8.1 Notation 265
8.2 Existence and Uniqueness Theorem 265
8.3 Remarks on the Application of the General Theory of Section 1 270
8.4 Additional Regularity Results 275
8.5 Parabolic Regularization; Direct Method and Application . . . 280
9. Equations of the Second Order in t; Transposition 283
9.1 Adjoint Isomorphism 283
9.2 Transposition 283
9.3 Choice of L 284
9.4 Trace Theorem 285
9.5 Variant; Direct Method 287
9.6 Examples 292
10. Schroedinger Type Equations 299
10.1 Notation 299
10.2 Existence and Uniqueness Theorem 299
11. Schroedinger Type Equations; Transposition 302
11.1 Adjoint Isomorphism 302
11.2 Transposition of (11.5) 303
11.3 Choice of L 303
12. Comments 304
13. Problems 307
Bibliography 309

Contents of Volume II
Chapter 4 Parabolic Evolution Operators. Hilbert Theory
Chapter 5 Hyperbolic Evolution Operators, of Petrowski and of Schroedinger.
Hilbert Theory
Chapter 6 Applications to Optimal Control Problems
Appendix Boundary Value Problems and Operator Extensions
Contents of Volume III
Chapter 7 Scalar and Vector Ultra-Distributions
Chapter 8 Elliptic Boundary Value Problems in Spaces of Distributions and
Ultra-Distributions .
Chapter 9 Evolution Equations in Spaces of Distributions and
Ultra-Distributions
Chapter 10 Parabolic Boundary Value Problems in Spaces of Ultra-Distributions
Chapter 11 Evolution Equations of the Second Order in / and Schroedinger
Type Equations
Appendix Calculus of Variations in Gevrey Type Spaces

Chapter 1
Hilbert Theory of Trace and Interpolation Spaces
The aim of this chapter is to give the fundamental results of the
theory of trace and interpolation spaces in the Hilbert case. Some
indications about the possible generalizations to the non-Hilbert case are
given in the Comments and in Section 14 along with the basic literature.
Sections 10, 12 (except 12.1), 13—15, although used in the sequel, may
be skipped on first reading.
1. Some Function Spaces
1.1 Sobolev Spaces
Let Q be an arbitrary open set in RM; x = {xlf . . ., xn} e Q, dx
= dxlt..., dxn. We denote by L2 (Q) the space of (classes of) functions u
which are square integrable on Q, i.e. measurable and such that
(l.i) IMIl>(0) = ( / M2<**) <«.
We shall often set
(1.2) L2(Q) = H°{Q).
It is a classical result that L2 (Q) is a Hilbert space for the scalar product
(u>v)L2(D) = j u(x)v(x)dx
associated to the norm (1.1). D1
Let m be an integer ^ 1. In short, the Sobolev space Hm (Q) of order m
on Q is defined by
(1.3) Hm(Q) ={u\D"ueL2(Q)V<x,\<x\ ^m}f
where
^ai + ■••+«„
°"= dx? a<-' a = K'•••>*<•}> 1*1 =*i+ ••• + *»•
1 The symbol □ will be used throughout this text to indicate the end of a
"logical unit".

2 1. Some Function Spaces
It must be stated precisely in what sense D"u in definition (1.3) is
taken.
For this purpose, we briefly recall the definition of distributions
on Q (see Schwartz [1]).
We define
(1.4) &}{Q) = {cp | <p infinitely differentiate on Q and with compact
support in Q}.
If K is a compact set in Q, we set
2K(Q) ={<p\(pe9{Q), (p with support in K).
With the norms
Pj(<p) = sup|Z)>(*)|, / = 0, 1,...
xeK
2K(Q) is a Frechet-space (i.e. metrisable and complete); then if Kn is
an increasing sequence of compact sets belonging to Q and whose union
is Q, we have algebraically
(1.5) ®(Q) = [J®Kn(Q)
and we provide 2{Q) with the corresponding inductive limit topology (i.e.
the finest locally convex topology which makes the injections 2Kn{Q) -»
-*@(Q) continuous; see Schwartz [1], Horvath [1], in particular
p. 165 and p. 171, where the explicit definition of a fundamental system
of neighborhoods of zero is given). D
Remark 1.1. Other spaces of infinitely differentiable functions will be
introduced in the following volumes. D
-We define
(1.6) 2'(Q) = dual of &(Q) = space of distributions on Q
and provide 2' (Q) with the strong dual topology.
We refer the reader to Schwartz [1] for structure theorems pertaining
to9'(Q). a
Remark on the notation.
If T eS>' (Q) and q>eSf{Q)t the value of T at <p will be denoted by
If 9? = complex conjugate of q>, we shall write
(T,<p) = (T,<p).
If TeL2(Q), (T,(p) coincides with [Ttq))L2ia).
Throughout this book, <#,/?> denotes a bilinear couple, so that
(jx, /5> is a sesquilinear couple (linear in oc, antilinear in /?), sesquilinear

1.1 Sobolev Spaces 3
couples being generally denoted by (<x.,(t) (with the particular notation
[*,|8] in Chapter 3). D
If T e2' (ii), its derivative d Tjdxj is defined by
»7) (■^■")--(r-^> *"«»■
which yields a linear continuous mapping
„ dT
1 -*
dxj
of 2'{Qy-> 2'(Q). Of course, D«T is defined by iteration.
We are now in a position to give a precise statement of definition
(1.3), if we note that
(1.8) 9{Q) cL2(£) c^'(£)
by identifying (which is permissible) every element ueL2 (Q) with the
distribution
<p->(u,<p).
Then, the derivatives Dx u in (1.3) are taken in the sense of distributions
on Q.
We provide Hm(Q) with the norm:
(1.9) IMIh«o» = ( I ll£aHl£W)1/2
\\*\*m J
and obtain
Theorem 1.1. With the norm (1.9), Hm(Q) is a Hilbert space, the scalar
product of two elements u,v eHm(Q) being given by
(1.10) (u,v)Hmio)= X {D*u,D*v)w
Proof. It is sufficient to verify that Hm(Q) is complete in the norm
(1.9). Let uk be a Cauchy sequence for this norm. It follows that, for
every oc with \oc\ ^m, D*uk is a Cauchy sequence in L2(Q). Since
L2(Q) is complete, we have
£"«*-> Y>« in L2(Q) V*, \oc\^m.
Set ip0 = u; since uk -» u in L2(Q), we have, in particular, uk -» u in
3d' (Q) and since the derivative operator is continuous in the sense of
@'(Q) we have
D«uk->D"u in &(Q).
Therefore y)a = D"u and u e Hm(Q). D
Remark 1.2. If mx > m > 0, we have the strict inclusions
Hmi (Q) c Hm (Q) c L2 (0) = #° (0). D

4 1. Some Function Spaces
1.2 The Case of the Entire Space
In the particular case Q = Rn, it is possible to give — and this
is important for the sequel — an equivalent definition of Hm(Q), by
making use of the Fourier transform. If u g L2 (Rn), the Fourier
transform u in L2(Rn) is defined by
1 f
«M = (2 )n/2 exp(-ixy)u(x)dx,
R"
xy = x1y1 + -" + xHyH9
the integral converging in the sense of L2 and u -► u is an isomorphism
of L2(R") onto L2(Rn). We set
u = IF u
and
u=^u = {2n)n/2 exp(ixy)u(y)dy.
R"
The Fourier transform extends by continuity to the space SP' of
Schwartz's tempered distributions, whose definition we now recall.
First of all, we define
& = {u | x* Dpu g L2 (Rn) Vol and V£}
(where x" = x\ . . . <n).
With the sequence of semi-norms
^-HK£^||L2(Rn),
£f is a Frechet space; of course every u e£f is (a.e. equal to a function)
infinitely differentiable in Rn and every u e Sf is rapidly decreasing at
infinity:
Va,Vj8, |*|" £*«(*) -> 0 if |a:| -> oo
{equivalent property to the above definition).
We easily verify that
(^(Dau) =(iy)aFu VueST, V*
[D^{u) =^((-ixfu) Vue^, V0,
and therefore that & zS£ {&>',&>) ((1)). In the same way,
#e^(y;y) and & <F u = &Fu = u, VueSf.
Therefore IF is an isomorphism of SP onto itself, with inverse F.
((D) We recall that if 0 and W are two topological vector spaces, J?(@; W)
is defined to be the space of linear continuous mappings of 0 -> W.

1.2 The Case of the Entire Space
Because of the symmetry of the kernel
1
—— exp( — ix y)
of «^, we have
[{& u)vdx = [u{$rv)dx V«, i/e.^.
R" R"
Next, we define
^' = dual space of SP t with the strong dual topology
and we define
&, & e S£ (&"; SP) by transposition.
Thus, Vwey, we have
where the brackets denote the duality between ^' and £P.
The formulas (1.11) are still valid "iuef?'.
Theorem 1.2. If Q = R", #m(R") way 6e defined by (1.3) or fry
(1.12) Hm(Rn) ={u\ue&",(l + \y\2)ml2 u e L2(Rn)}
(where \y\2 = y\ + • • • + y2n)y
the norm
(i.i3) IHIh-kr.., = ll(i + lyl2)m/2^ll^(R»>
being equivalent to the norm (1.9).
Proof. From (1.11) and Plancherel's theorem
II^HImR") = l|y^llL2(R"),
so that (1.9) yields (for Q = R")
(1.14) IMIh~(r»)= f ( I y2*)\ti(y)\2dy-
But for a suitable constant C:
(i + \y\2)m^ Z y2a^c(i + |y|2)-,
which, together with (1.13), (1.14) yields
HI W|||Hm(Rn) = II U llHm(R") = ^ HI W|||ffm(R")' LJ
Remark 1.3. Q) (Rn) is dense in #m (Rn) (this is easily seen by regulariza-
tion and truncation), but this is not true in general: on the contrary
Sf(Q) is not generally dense in Hm(Q). We shall return to this point
later. D

6 1. Some Function Spaces
1.3 The Half-Space Case
In case Q is the half-space [x \ xn > 0}, we introduce another
equivalent definition of Hm(Q). D
The space L2(a,b;X).
Let X be a Hilbert space. L2(a,b; X) denotes the space of (classes
of) function /, strongly measurable on [a, b] with range in X (for the
Lebesgue measure dt on [a,b]) and such that
/ b \1/2
(1.15) M \\t(t)\\2xdt\ = II/IIl2(„,*;*)< +oo,
where || \\x is the Hilbert norm of X.
With the norm (1.15), L2 (a, b; X) is a Hilbert space (Bourbaki [1]). D
Distributions on ]a,b[ with range in X.
We recall that, if 0 and ¥ are two topological vector spaces, we have
set
(1.16) ££ (0; XP) = space of linear continuous mappings of 0 into !P
(the "adequate" topology on this space being defined in each particular
case).
We call (following Schwartz [6]) space of distributions on ]a, b[ with
range in X, denoted by @>'(]a, b[] X), the space
(1.17) 9'Qa,b[;X) = J?($(]a,b[); X),
provided with the topology of uniform convergence on the bounded
sets of 0(>,&[).
Therefore, if f e &'(]a, b[; X), then V<p e &>(]a,b[), </, <p} (value
of / at (p) is in X and q> -» </, <p) is a continuous mapping of @(]a, b[)
into X. df
The derivative — of /e ^' (]a, 6[; X) is defined as the unique
element of this space which satisfies
(U8) (^•")--(/'-&) ^^Q«'»d-
(the equality (1.18) takes place in the space X).
The mapping ,,
/- —
dt
is a continuous mapping of @'(]a,b[;X) into itself. □
Remark 1.4. The fact that X is a Hilbert space plays wo role in the
preceding notions on distributions taking their values in X; later on

1.3 The Half-Space Case 7
(in volume 3), we shall work with much more general notions. But for
the purposes of this chapter, the hypothesis that X is a Hilbert space
is sufficient. D
Now, if f e L2(a,b\ X), we define
fe&Qa,b[;X)
by
b
(1.19) <f,<p>= tf(t)<p(t)dt (eX) V<pe9Qa,b\),
a
so that we have a (continuous linear) mapping / -» / of L2(a,b\ X) -*
-* £&'(]a, b[; X). This mapping is one-to-one; and so we identify /
with / and obtain
(1.20) L2(a,b',2C)c9'Qa,b[',X).
Consequently:
Proposition 1.1. For f eL2(a,b; X), —, —T, . . . may be defined
dt dt
as distributions on ]a,b[ with range in X. D
Remark 1.5. The following may be verified as an exercise (as in
Theorem 1.1, but now using the continuity of the derivative mapping
for vector distributions). Define:
/I/, /U)=-r- f^=-P-€L2(a,b;X)\
dt dtm
Hm(a,b;X) =
with the scalar product
m r
Hm(a,b; X) is a Hilbert space. D
We are now ready to prove
Theorem 1.3. When Q = {x | xn > 0}, Hm{Q) may be defined by (1.3)
or by
(1.21) Hm{Q) = \u\ueL2(0, oo; H-'iRlr1)),...,
dxi dxn
and
2 £ II SJU I'
J=0\\ OX„ ||l,2(0. oo;H»>-J(R;, '))

8 1. Some Function Spaces
Proof. 1) If u eL2(0, oo; i/m(R"r1)), then —% is defined in the
sense of «
^'(]0,oo[;//m(R!;r1))
and therefore the conditions "—\e L2(0, oo; Hm~J(R"^1))" have
meaning. J
2) If ueL2(0, oo;i/m(R71)), then since, for all \oc\ ^ m, D"X, is
a continuous linear mapping of i/m(R","1) into ^(R","1), we have
(1.22) Z)>eL2(0, ooi/Z^R^r1)) V|*| ^ w
and since by Fubini's theorem
(1.23) L2(0, oo;HO(R"x71)) = L2(Q),
we have
(1.24) D$,ueL2(Q) V*, |*| ^ m.
Conversely, if u satisfies (1.24), then u satisfies (1.22); therefore,
for almost all xn, u(-,xH) Gi/m(R"r1) and
00
\ \\u(.fxn)\\2HmiRn:i)dxn= £ \\(D«x,u)\2dx<oo,
o \«\*"£
so that ueL2(0, oo; i/m(R","1)) (since u is measurable and takes its
values in i/m(R^"1)).
Therefore, the fact that u e L2(0, oo; i/m(R"r1)) is equivalent
to (1.24).
3) Consequently
dJu , S-^
—-eL^Coojff-^ier1)) V/oDJ-TGl2(fi) V|*|£m-/,
V/oZ)a«eL2(£) V|*| ^ m,
from which the theorem follows. D
1.4 Orientation
Property (1.21) justifies the introduction of the notions of the next
section, these notions playing an essential role in the following chapters.
This will bring us to:
O trace theorems and "intermediate spaces",
O interpolation theory.

2.1 Intermediate Spaces
9
2. Intermediate Derivatives Theorem
2.1 Intermediate Spaces
Let X and Y be two Hilbert spaces which, in order to slightly
simplify our account (and also because this will suffice for our purposes),
we suppose to be separable with
(2.1) X a Y, X dense in Y with continuous injection. □
Remark 2.1. The space Hm(Q), introduced in (1.3), is separable.
Indeed, via the mapping
u->{D"u\ |*| ^ m]
it can be identified with a closed vector subspace of a product of L2 (Q),
this product space being separable because L2(Q) is. □
Let ( , )A and ( , )Y be the scalar products in X and Y respectively.
The operator A.
The space X may be defined as the domain of an operator A, which
is self-adjoint, positive and unbounded in Y (in fact, A is not unique!),
the norm in X being equivalent to the norm of the graph
(2.2) (||«||* + \\AufYyi\ ueD(A) = X.
The result is classical (see Riesz-Nagy [1]). Let us briefly recall a
procedure (which, as a matter of fact, is linked to the variational
formulation of elliptic boundary value problems, Chapter 2, Section 9).
We denote by D(S) the set of u's such that the antilinear form
(2.3) i/-> (u, v)x, veX
is continuous in the topology induced by Y. Then
(2.4) («,«)* = (Su,v)Y,
which defines 5 as an unbounded operator in Y, with domain D(S).
It is easy to see that:
D(S) is dense in Y,
S is a self-adjoint operator, i.e. 5 coincides with 5* = adjoint of 5
(which also means that their domains coincide) and 5 is strictly positive]
indeed,
(Sv,v)Y = \\v\\x ^ (constant) \\v\\Y.
(For the definition and the principal notions relative to self-adjoint
operators we refer the reader to Stone [1], Riesz-Nagy [1], Yosida [2],
among others). Using the spectral decomposition of self-adjoint operators,
the powers Sd of 5, 6 g R (or even fleC), may be defined (see the texts
just cited and the reminders given in Section 2.3, below).

10 2. Intermediate Derivatives Theorem
In particular, we shall use
(2.5) A = S112.
The operator A is self-adjoint and positive in Y, with domain X.
From (2.4), (2.5) we deduce
(2.6) (u,v)x = (Au,Av)Y Vw^el. □
Remark 2.2. The operator 5 depends on the choice of the scalar
products on X and Y (of course, without changing the topology of X
and Y) and therefore A also depends on these scalar products; thus,
it is not intrinsically linked to the spaces X and Y. D
We give the following definition of the intermediate spaces [X, Y]e:
Definition 2.1. Under hypothesis (2.1) and with A defined by (2.5),
we set
(2.7) [X, Y]e = D(A1-6), (domain of A1'6), 0 ^ 6 ^ 1,
with
(2.8) norm on [X, Y]e = norm of the graph of A1'6, i.e.
(IMlS + M1-'^)1'2- Q
From the properties of the spectral decomposition we immediately
have that X is dense in [X, Y]e. D
Remark 2.3. According to Remark 2.2, it is not obvious that the
space [X, Y]e is intrinsically linked to X and Y; this will be shown in
theorems 3.2 and 4.2, below. We shall obtain:
if Ax and A2 are two positive, self-adjoint operators in Y, with
domain X, then
D(A\~e) = D{Al2~6), with equivalent norms. D
Remark 2.4. We have:
[X, Y]0 = X
[X, Y]x = Y. D
2.2 Density and Extension Theorems
The space W(a,b).
Let a and b be two real numbers, finite or not, a < b.
Let X, Y be Hilbert spaces as in Section 2.1, and m an integer ^ 1.
We set
(2.9) W(a,b) = L|«eZ,2(tf,&;X),—- = u™eL2(a,b; Y)

2.2 Density and Extension Theorems 11
where w(m) is taken in the sense of distributions in 3i' (]a, 6[; X). Provided
with the norm
(2-10) || u y^n = [|| u fLHa>b.X) + || «<•> l^^.-n]1'2
W(a, 6) is a Hilbert space (it is complete in the norm (2.10) because of
the continuity of differentiation in t in the sense of S'(]a, 6[;X)).
TAe s^ace 0(|>,&];X).
We denote by 2{\a, b]; X) the space of functions which are infinitely
differentiable for a ^ t £ b, with range in X and of compact support.
There are three cases:
1) a = — oo, b = +oo; then, the space coincides with £&(R;X)]
2) a finite, b = +oo; then, the functions of Sf[[at b]\ X) vanish for
sufficiently large t (the case a = —co,b finite, follows by symmetry);
3) a and b finite.
Density theorem:
Theorem 2.1. The space @([atb]', X) is dense in W(a,b).
Proof. First case: a = —oo, b = + oo.
Let {£„} be a regularizing sequence:
( —oo
| +oo
| qn with support in [>„,£„], ocn>pn -» 0.
If WGl^(-oo, +oo) = W(R), then
— oo
u* Qn(t) = Qn(t — a) u(a) da -> u in L2 (- oo, + oo; X),
+ oo
(u * e„)(m) = w(m) * q„ -» w(m) in Z,2 (- oo, + oo; Y).
Therefore, it is sufficient to approximate (in the sense of W(a, b))
the functions v of the form
(2.11) v = u*y, q>e@(R).
by elements of 2{\a,b\\X).
We already have:
vik) = u * 99(fc) V&,
therefore
(2.12) v^eL2(-od, + oo;X) VA,

12 2. Intermediate Derivatives Theorem
so that, as is easily verified:
(2.13) v is an infinitely differentiable function of R -» X.
It is now sufficient to truncate. For instance, let ^e^(R), tp(t) = 1
for |*| <; 1, ip(t) = 0 for |;| ^ 2. Let ^ be defined by
(2-14) ^W=r(^)-
Then, because of (2.13):
(2.15) VNve9([a,b]]X).
We have:
ipNv-*v in Z,2( — oo, +oo; X)
and therefore, we shall obtain the desired result if
(2.16) (yN w)o»>-> w0»> in L2(-oo, + oo;X)
(therefore, in particular in L2(— oo, + oo; Y)).
But y>Nfl(m) -» ^(m) in L2(-oo, + oo;X) and (2.16) follows from
y{*> wo»-*>-> 0 in L2(-oo, +oo;X), iV-» oo, £ ^ 1;
but this last statement is an immediate consequence of the definition
(2.14) of yN. □
Second case: a finite, b = +oo.
Of course, the situation is invariant by translation and we may
assume a = 0.
Let u e W(0, oo); for A > 0, we define
(2.17) uh(t) = u(t + h), t > 0 (which has meaning e.a.).
Since
u(hm)(t) = «<"■>(* + h) a.e. for * > 0,
we see that
(2.18) uh->u in W(0, oo) as h -» 0.
Therefore, it is sufficient to approach % (in the sense of W(0, oo))
with elements of 2{\0, oo];X), with A //#erf.
Consider a scalar function 0, intinitely differentiable on R, &(t) = 1
if t ^ -A/2, 0(0 = 0 if t£ -h, and let
0(t)u(t + h) if 2 ;> -h
0 if * < -h.
v(t) =
Then
v = uh a.e. on 2 > 0 (we did what was necessary for this)
and
V 6 W(—QO, + 00).

2.2 Density and Extension Theorems 13
From the first case, there exist fne@(R',X), with
fn-* v in W( — co, +00).
If gn = restriction of /„ to [0, + oo[, we have:
gn -» (restriction of v) = uh in W(0, +00),
gn e 9([a,by,X);
hence the desired result. D
Third case: a and b finite.
Let oc and /} be two scalar functions, with the properties
<*,Pe®([a,b)], oc(t) + P(t) = 1 for a^t^b,
oc (resp. /?) vanishing in the neighbourhood of b (resp. a).
Then every u eW(a,b) may be written
U = OCU + (} U
and if we define
/—' [«w for fl^/^i >—' \(iu for a^t^b,
10 for t > b [0 for t < a,
we have, thanks to (2.19):
ocu e W(a, +co), P u eW( — 00, b).
Therefore, following the second case, there exist sequences
/„ (resp. gH)e9([a,+co]',X) (resp. 9 ([ - 00, b]; X))
such that
/„ -* ocu (resp. gn -* /} u) in W(a, +00) (resp.W (— co, b))
and by restriction to [a, b] (as at the end of the second case) the desired
result follows. □
Remark 2.5. It is possible to reason directly on the third case by
"extending" the definition of the functions by homothetic mappings
(instead of the translations used in the second case).
Extension theorem:
Theorem 2.2. It is assumed that at least one of a or b is finite. There
exists a continuous linear operator u -* p(u) from
W(a,b) -> W(-ao9 +00)
such that
(2.20) p(u) = u a.e. on ]a, b[.
(2.19)

14 2. Intermediate Derivates Theorem
Proof. By the method of the " third case " in the proof of Theorem 2.1,
we are brought back to the case [a, +oo] (the case [— oo, b] of course
being analogous by a change of t) and we may assume a = 0. Therefore,
let us define p (u) for the case [0, + oo]. We use the method of " extension
by reflection".
For we®([0, oo];X), we define p(u) by
(2.21) P(u)(t)=\
[u(t) if t> 0,
m
£**«(-& 2) if t < 0,
where the numbers <xk are defined by the conditions
(2.22) —7p(u)(0) = «a)(0), 0^/^m-l, V« e#([0, oo];X),
i.e.
m
(2.23) £(-l),/*/** = 1, O^'^-l
(the #fc's are well-defined by this system).
The function p(u) is in W( — oo, +oo), thanks to (2.22), and
p{u)™(t) =
«(m)(2) * > 0,
m
fc=i
so that
(2.24) ll/>Wlk(-oo. + oo) ^ c IIHkco.oo) (c = constant)
is easy to verify, and therefore (from Theorem 2.1) the mapping u->p(u)
extends by continuity to a linear mapping, still denoted u -> p(u), of
W(0, oo)-» W(-oo, +oo).
The property: p(u) = u on t> 0 (which is satisfied for u regular) yields
p(u) =u a.e. for t > 0 by passage to the limit (and for u e W(0, oo),
p(u) is again defined by (2.21), this time a.e. in t). D
2.3 Intermediate Derivatives Theorem
First, we recall the concept of measurable hilbertian sums.
On [A0, +oo[, let there be given a Radon measure dfx{X) ^ 0. For
each A e [A0, + oo[, let t) (A) be a Hilbert space on C (for which the scalar
product and the norm are denoted ( , )l)U) and || ||^a), respectively).

2.3 Intermediate Derivatives Theorem 15
The spaces i)(A) are said to form a [x-measurable field if the I) (A)
"depend //-measurably on A", which means: define a family Jl of
functions:
A-WW) of [A0,+oo[->f)W
having the following properties:
(i) V/e*#, the function A -» H/Wllgu) is //-measurable;
,..v ( if g is a function with values in I) (A) such that, V/ e *#,
[ A -> (/(A), g(A))^a) is //-measurable, then geJi\
..... j there exists a sequence /x .../„... of elements of .^ such that,
I VAe[A0, + oo[, the sequence /i(A),...,/n(A),... generates i)(A).
The elements of ^# are the measurable functions taking their values
in W).
r^^ 2/^ space:
e
^'s defined as follows. A measurable function A -> /(A) is m ^ if and only if
+ 00
11/11? = J ll/WII^)^W<cx).
Ao
For f,geij, their sca/ar product is defined by
+ 00
(/.*)* = / (/W,g(%A)^(A).
Ao
It can be shown (Dixmier [1], p. 146; see also Gelfand-Vilenkin [1],
p. 114 of the English edition) that the f) so defined is a Hilbert space.
The space i) is called the measurable hilbertian sum (or the direct hilbertian
integral) of the spaces I)(A).
We shall make use of this notion in the proof of the next result;
a result upon which we shall frequently call in this book (under the
name of "intermediate derivatives theorem").
Theorem 2.3. Let X, Y be two Hilbert spaces with the properties (2.1)
and let [X, Y]e be defined by (2.7). If ueW(a,b) (defined by (2.9)),
then
(2.25) ««> e L2 (a, b; [X, Y]J/m), 1 ^ j£ m - 1
(for; = 0 and m, (2.25) still holds and reduces, according to Remark2.4,
to the definition of W(a, b)). Furthermore, u -» uU) is a continuous linear
mapping of
W(a,b)->L*(a,b;[X,Y]Jlm).

16 2. Intermediate Derivates Theorem
Proof. 1) Thanks to the extension operator p of Theorem 2.2, it
is sufficient to prove the theorem for a = —oo, b = + oo. We may
use the Fourier transform in t:
u-*u=&ru, u(r) = ,— exp( —i/r)«(/)
V27T .^
<*/
which is a unitary isomorphism of L2(Rt; £) -*• L2(RT; £) whenever £
is a Hilbert space.
Following Schwartz [6], we define the space S?'(E) of tempered
distributions with values in E by
ST'(E) =y'(RtiE) =£>(<?; E);
every «£/(£) may be written, in non-unique fashion, as a finite
sum of derivatives in t of functions of the form (1 4- \t\)k fk, fk e L2(E),
where the derivatives are given by
dmu (dmw\
dtm \ dtm m
For «e «?"(£), we define &ueSf'(E) by
which again defines an isomorphism of Sf' (E) onto itself. We still have
the "usual" rules, in particular
I d3u\
(2.26) y\T) = (iT) V«e .$"(£).
Consequently:
UgL2(Rt;X)
(2.27) «eW(-oo, +oo) o
It-^gL2(Rt;Y)
and the properties (2.25) to be demonstrated are equivalent to
(2.28) T*ueL2(Rt;[X,Y]Jlm).
2) We now use the diagonalization of A. According to the theory of
spectral decomposition (see Dixmier [1], and Gelfand-Vilenkin [1]),
there exist:
e
(i) a measurable hilbertian sum i) = j I) (A) dju(K),0 < A0 ^ A < oo,
d^(A) = Radon measure ^ 0 on [A0, +oo[,
(ii) a unitary operator °U of Y onto 1) which maps X onto ij1, where
(2.29) lh ={w|we^lAwe^}l

2.3 Intermediate Derivatives Theorem 17
with
IMI*t = M PWHDWwdrW
1/2
and such that
(2.30) V(Au)=A(Vu) VueD{A) = X. D
We verify right away that
(2.31) % is an isomorphism of [X, Y]e onto I)i-e,
where
(2.32) ^-•={»|fe^AM!/6^} 0^0 g 1.
with
(+oo \l/2
j x2ii-9}\\vmla>*i*w) .
We note that v e f)i_e if and only if A1""0 vet) (since A0 > 0 and
1 — 0^0), and the norm on i}x _e is equivalent to the norm of the graph.
3) For /eI2(R; Y), we define <% f by
(«/)W = *(/W) a.e.
and similarly for feL2(R\ [X; Y]e). Then
(2.33) ^ is an isomorphism of L2(R; [X, Y]e) onto L2(R; f)!_e).
With the notations of (2.27), we also have:
\^ueL2{Rx\^)
(2.34) ueWl-oo, + oo)~{
[Tm<%ueL2{Rx;fy.
We set v = °U w(= v(A, r)). Then conditions (2.34) may be written
+ 00 +00
(2.35) j j (X2 + r2m) HA.T)||Jtt)<ty(AMT < +«.
— oo Ao
z#/wcA is equivalent to
(2.36) (A + \r\m)v(?i,r)eL2(Rx'>ij).
Still with the same notation and using (2.28) and (2.33), we see
that the property to be demonstrated is equivalent to
(2.37) X1--»m\T\Jv(X,T)eL2(Rt\fi),
with
(2.38) ||A1"-"-|tK»(A.t)|L2(r,.,) ^ C\ ||(A + ItD^A.t)!^^^,.

18 2. Intermediate Derivates Theorem
But these last properties result from the inequality:
A1-""'|t|'£C2(A + |tD
[apply the inequality
fr-Jlm |TL/ < J_Xi-J/m)p + J_ \r\Jp\
— +— =1, with (l_-Lu = if jp>=m\ Q
p p' \ mj ]
Remark 2.6. From (2.31) and (2.32), we have that X is dense in
[X;Y]e V0. D
2.4 A Simple Example
We introduce a situation which we shall often meet in later chapters.
Let V and H be two Hilbert spaces, V a H, V dense in H with
continuous injection.
Identifying H with its anti-dual and V denoting the anti-dual of V
(by abuse of language, we adopt the same notation for the dual and anti-
dual, in order to prevent meaningless distinctions in the practical
examples), we have:
(2.39) VcHcV,
each space being dense in the following one.
If (u, v) (resp. ((«, v))) is a scalar product on H (resp. V), we define
the operator A as 5 in Section 2.1 by taking V = X and H = Y. Then
(2.40) D(A) c V c H a V, V = D(A
l/2\
We diagonalize the operator A (see the proof of Theorem 2.3) by a
measurable sum 1) and a unitary operator °ti of H onto fj and D (A) onto
fjj ={v\vef),Avefy = {v\2.v el)}.
Then ^ is an isomorphism of V onto lj1/2 and of F' onto Jj_1/2, where
We immediately obtain:
Proposition 2.1. With the notations (2.39), (2.40), we have
(2.41) [7, 7']1/a = H,
(2.42) [W).#]i/2 = P-

3.1 Continuity Properties of Elements of W(a, b) 19
In particular, the intermediate derivatives theorem yields:
Proposition 2.2. // u el2(0, oo; V) and u" el2(0, oo; V), then
u' eL2(0, oo;#). D
2.5 Interpolation Inequality
The following inequality is an immediate consequence of the
definition of the spaces [X, Y]e and the fact that
Proposition 2.3. For every u gX,
(2.43) Mu.ru ^C Mi'9 Mr- D
3. Trace Theorem
3.1 Continuity Properties of Elements of W(a9 b)
Generally, if E is a Hilbert space, we set
C°([a, b])E) = continuous functions of [a, b] -+E
if a and b are finite;
Continuous bounded functions ot t ^ a -» E if a
is finite and b = + oo;
continuous bounded functions of R -» E if
a = —oo, b = +oo.
We provide 3#{a,b',E) with the norm
ll9>ll*<a.&;E) = SUP IIpWIIe-
te[a,b]
Theorem 3.1. With the notation (3.1), for ueW(a, b) we have:
(3.2) ««> e S (a, b; [X} Y]„+1/2)/m), O^^m-1,
w -► ^a) fomg a continuous and linear mapping of W(a, b)
-«(«,&;[X,Y]0+1/2)/j.
Remark 3.1. We have:
[X, F];/m c [X, Y]u+
l/2)/m-
Therefore, if u EW(a,b), then wU) is square integrable with values in
[X, ¥];/„, and continuous with values in a Ztfrger s/>ace. D
Remark 3.2. We must make (3.2) somewhat more precise: the
function uu\ which is known to belong to L2(a,b; [X, Y]J/m) (intermediate
derivatives theorem) is, after a possible modification on a set of measure
(3.1) a(a,b;E) =

20 3. Trace Theorem
zero, a continuous mapping of
[«,&]-»[x,Y]0+1/2)/.
(with the modifications appearing in (3.1) if a and b are infinite).
In this form, the last part of the theorem is slightly ambiguous
(the set on which uU) is modified could depend on u); a more precise
statement of the theorem is (see Theorem 2.1):
[ the mapping u -» uU) of 2'([a, b]\ X) -» 2'([a, 6]; X) extends by
(3.3) ' continuity to a mapping of
( W(a,b)->®(a,b;lX,Y]u+l/2)/m)
Proof. 1) As in the proof of Theorem 2.3, we are led back (with the
help of Theorem 2.2) to the case a = —oo, b = + oo. Again we use
the Fourier transform in t and the diagonalization °li. Therefore, for
ue2(R',X), let
v = <2f«.
We show that
(3.4) \\uU)h(a,b;txna+i/2)/m) ^ c\\u\\Wiatb),
which proves the theorem.
Now +00
qi uU)(t0) = f exp(27r i t0 x) [2n i r)J v(r) dx, t0 e R,
— oo
and (3.4) will follow from
(3.5) \\®u^(t0)\\^_a+i/2)/m ^ C\\w\\LHRx;», w = {X+ \tF)v.
2) We have «,
Ao
oo / +oo
oo / +oo \ / +oo
Zc^f-^dpwl j Jll^jM \\w(X,r)\\^dr]
0 \-oo ' \-oo /
+ oo +00+00
= (sett = ^"(7) =c, J (y^^)2 ^<t frf^ (A) f II ^(A, r) ||,2(A) rfr
Ao
= c2 \ \ \\w\\la)d*d!*W>
Ao — oo
whence (3.5).

3.2 Trace Theorem 21
3.2 Trace Theorem
We state the result for a = 0, b = 4- oo (as will be the case for most
of the applications and with no loss of generality!):
Theorem 3.2. Let u e W(0, oo). According to Theorem 3.1 \
u^(0)e[X,Y]a+1/2)/m, Og/^m-1.
The mapping
(3.6)«^{«<'>(0)|0£j£m-l} of ^(0,oo)->nCX,Y]a+1/2)/m
j=o
is surjective.
Proof. 1) We start with a very general remark which is independent
of the Hilbert structure of the spaces:
(3.7)
to show the surjectivity of (3.6) it is sufficient to verify the
surjectivity of the mapping u -* uJ(0) of 1^(0, oo) -*•
-* [X, Y](j+i/2)/m> f°r arbitrary fixed j.
<*) ~ i ° if * * /' ° = * = m - l
Indeed, assume the surjectivity for arbitrary fixed /.
m-l
Let {aj} e f] [X, Y]u+1/2)/m. Then, there exists us e W(0, oo) with
We construct Uj e W(0, oo), with
(3.8) ^'(0) = , .
( aj if ft = j.
Then
m-l
U = £ Uj e 1^(0, oo) and tf'(0) = a, V/, 0 ^ / ^ w - 1;
whence the desired surjectivity — after the construction of Uj.
For this purpose, we define
(3.9) [/.(<) = £Cr. Uj(rt),
r=l
with
[0 if ft 4= /, 0 ^ ft ^ w -1
(3.10) I^fccr,=
1 if ft = /
(which properly defines the cr/s); of course, we have UjeW(0, oo)
with (3.8), since Uj -» Uj is a continuous mapping of W(0, oo) into
itself.

22 3. Trace Theorem
2) Therefore, there remains to show the surjectivity with fixed j.
Using the diagonalization (as in Theorem 2.3), we consider
a (A) = ae^i-u+i/D/m-
We shall construct w to satisfy
weL2(0, oo;^),
(3.11) |^(m)eL2(0, oo;fj),
ie^>(0) = a,
(3-12) IMI^O.oo^) + ll^llwo..;,) ^ C NII^.(, + 1/2)/m.
(Then u(t) = <%-l(w{t)) will yield ueW(09 oo) with ^(O) = a and
the desired result will follow).
For the construction of w, we consider a function cp e £?([0, +oo[),
with
^(O) = 1
and we set
(3.13) w(A,t) = A-1/ma(A)(p(A1/mt).
Then, the properties (3.11) and (3.12) may be verified by an elementary
calculation. D
Remark 3.3. The proof of the theorem shows that there exists a
continuous linear right inverse 0t
(3.14) {aft:* ®uoi Y[\x> Y]u+i,2ym - ^(0, oo)
J = 0
such that
(3.15) u<» (0) = aJt 0 ^ / ^ m - 1. □
Remark 3.4. The "intermediate spaces" [X, Y](J+i/2)/m appear as
the spaces described by the "traces" uU)(0); for this reason, the spaces
[X, Y](j-+i/2)/m are often called "trace spaces". D
Remark 3.5. Since u -» u(j) (0) is a continuous mapping of W(0, oo) -*•
-> K y]a+i/2)/m. we have
(3.16) ||u^(0)\\lXtY,(J+l/2)/m < Cj(\\u\\L2(0t„;X) + \\u™\\LH0tO0;Y)),
for a suitable constant Cj.
But, with q denoting a positive parameter, set
««W = q~J u(qt)>

4.1 Orientation. Definitions 23
and apply (3.16) to uq\ since ^(O) = uU)(0), we have
(3-17) ll«('>(0)IUn(J+1/2)/m
for arbitrary q. Choose q such that (u being fixed)
?-'-1/2 IMWoo;*) = r~J-112 N(m)||L2(0.oo;y)-
Then (3.17) yields
(3.18) ||^>(0)||[x,y](,+1/2)/m ^ 2c, \\u\\Ui+JJxT H-'ll^to/x/?,. D
Remark3.6. Since the mapping ft-* w(-/) (0) is a surjection of W(0,oo) -»
-*• [X, y](J-+i/2)/iii» it is natural to introduce the "quotient-norm"
(3-19) \Ha.y^l/2)/m= JnlJMmo.x).
(MO)(0)=a
On [X} Y]o+i/2)/m.
77&ew, 2Ae norm (3.19) is equivalent to the norm (2.8). Indeed, provided
with the norm (3.19), [X, Y]u+1/2)/m is the quotient of the Hilbert
space W (0, oo) by the closed subspace of functions cp such that cpU) (0) = 0
and is, therefore, also a Hilbert space. Since u -» «U) (0) is a continuous
mapping of W(0, oo) -» [X, Y](j.+ 1/2)/m, we nave (3.16) and therefore
Wahx.Yia+l/iUm ^ ^ua)^=Jlwll^(o.oo) = cj \\\a\iix,n0+1/2)/m>
so that the norms (2.8) and (3.19) are equivalent (closed graph theorem;
we could also use the right inverse (3.14)). D
Remark 3.7. Remark 3.5 shows that the space [X, Y]u+1/2)/m, with
its topology, is intrinsically linked to X and Y (see Remark 2.3). For
[X,Y]e, with arbitrary 0e]O, 1[, the analogous question will be
answered in Sections 4 and 10, below. □
4. Trace Spaces and Non-Integer Order Derivatives
4.1 Orientation. Definitions
An analysis of the proofs of Theorems 2.3 and 3.1 easily shows that,
especially when a = — co, b = +co, the hypotheses "integer m" and
"integer /" do not intervene in an essential manner.
For instance, (2.27) leads to the definition (we replace m by s to
avoid any confusion):
(4.1) W(-co, + oo;s;X, Y) = {u \ u e!2(Rt; X), \r\su eI2(Rt; Y)}

24 4. Trace Spaces and Non-Integer Order Derivatives
for arbitrary s > 0, integer or not. Provided with the norm
(4.2) {ml**,-.* + IIMs«||£2(R,;r))1/2 = ll«llir(-..+.;.;x.n.
W (— oo, + oo; s; X, Y) is a Hilbert space.
Of course, W( — co, + oo; m\ X, Y) = W( — oo, + oo), with the
notation of Section 2. □
Remark 4.1. We shall often say that the derivative of order s of u
is in L2(RT;Y). D
The subspace of functions u such that Dktu eL2(R; X) V& is dense
in W( — oo, +co;s;X, Y) (immediate, by regularization). D
We shall now extend Theorems 2.3 and 3.2 to this setting.
4.2 "Intermediate Derivatives" and Trace Theorems
Theorem 4.1 ("intermediate derivatives")* For every
ueW(-oo, + oo;s;X, Y),
we have
(4.3) \Tl'ueL2(Rz;[X,Y]rls), OZr^s,
and
(4-4) \\\r\ru\\mK,;ix,Yjris^ c \\u\\m
-oo, +co;s;X,Y)'
Proof. In the same way as for the proof of Theorem 2.3, we are led
to verify (compare with (2.38)) that
ji-r/i|T|r g constant (A + |r|s),
which is immediate (in fact, simply let / = r, m = s in the proof of
Theorem 2.3!). D
Theorem 4.2 (traces). For every
ueW(-oo, +oo;s;X, Y)
we Aave
(4.5) «w (0) e [X, Y]0+1/2)/1, 0 £ / < s - *.
Furthermore, the mapping
(4.6) «^{«a)(0)}o^<s-i/2
W(-oo,+oo; s;X,y)^ [1 C^.y]o+i/2)/.
0£j<s-l/2
is surjective.
Proof. 1) In fact, we have (which makes (4.5) more precise):
(4.7) «<'>e0(-oo, +oo;rx,Y](J+1/2)/s).

4.2 "Intermediate Derivatives" and Trace Theorems 25
By the method of the proof of Theorem 3.1, this brings us to the
necessity of verifying that
+ oo
|T|J^l-U+l/2)/s\2
i(J
Ms + A
dr < constant.
But setting r = 2.1/sa, this integral becomes
+ oo
' Iff!
J(-
cr|s+ 1
J \2
da,
which is finite if (and only if) ; < s — \.
2) The same procedure as in the proof of Theorem 3.2 (valid on
( — 00, +oo) as well as on (0, +oo)) shows that it is sufficient to prove
that u -*> uJ(0) is a surjection for fixed j (note that "the homothetic
mapping ": u -» u (r.) is a continuous mapping of W (— oo, + oo; s; X, Y)
into itself).
As right inverse, we take (compare with (3.13))
(4.8) w(JL9t) = X-Jlsa{X)(p{Xllst).
We verify that w eW( — oo, + oo; s; -X", Y) and in particular that
(4.9) |r|^(A,r)eL2(Rt;I)).
But
+ oo
1 f
w{X, x) = i— exp(-irt) w{X, t) dt
— oo
= A-(-/+1)/sa(A)^(A-1/sT)
+ oo +oo
= / ^2CU+1)/s]lkWII?a,^(A) / \x\**\<p{l-Vsx)YlT
and therefore
+ oo +oo
— oo Ao
Ao
+ oo
= / A-2[a+1)/fl««(A)ll£»<WM2+1/, j \<p(m2d£
Ao —oo
= (constant) ||;i1-(-/+1/2)/sa||£ < oo,
whence (4.9). □

26 4. Trace Spaces and Non-Integer Order Derivatives
Now, an obvious question is the following: consider again the
mapping
(4.10) u->uU)(0)
of ^(R; X) -> X. If j ^ s — \} is it possible to find a topology on X
such that (4.10) is continuous when ^(R; X) is provided with the
topology induced by W( — oo, + oo; s; X, Y)? The answer is wo:
Theorem 4.3. For any fixed f =)= 0 m X, 2/^ mapping
(4.11) «->(«(J)(0),f)z
o/ ^ (R; X) -> C *'s not continuous when Q) (R; X) is provided with the
topology induced by W (— oo, + oo; s; X, Y), if
(4.12) /£*-*•
Proof. By Fourier transform in t and if v = <% w and ?y = r\ (X) = °ll f,
we have
+ 00 +00
K"(0),f)X = / J (i t)'(»(A.T), *(%„*<*,! (A) <*T
Ao —oo
= ((\T\° + l)v(l,T),g(X,T))LHRt.f)),
where
Then, since \\u\\W(_O0t + a0.tS;XtY) *s equivalent to
||(|t|' + A)»(A,t)||W(r..6),
if (4.11) was continuous for the topology induced by
^(_oo, +oo;s;X, Y),
we would have
g(A,r)eL2(Rt;D),
which is false if (4.12) holds. □
Remark 4.2. It can be seen, as in Remarks 3.5 and 3.7, that
[X,Y]u+1/2)/s, 0^j<s — \, may be defined as the trace space
described by uU)(0) as u describes W( — oo, +oo;s;X, Y), with the
quotient norm:
(4-13) !H«.n(,+1/i)/. = J(n05=a ll«lk(-».+«»;.;x.y)- □
Since we are free to choose the parameter s, we see that [X, Y]d
and its topology are intrinsically linked to the couple {X, Y}. Another
demonstration of this property follows from Theorem 10.1, below. □

5.2 Interpolation of a Family of Operators 27
5. Interpolation Theorem
5.1 Main Theorem
Let {9Ct $/} be a second couple of Hilbert spaces having properties
analogous to the couple {X, Y}.
Let n be a continuous linear operator of Y into <3f and of X into #";
in short
(5.1) neX(X\X)c\£e{Y\<8/).
Then, we have
Theorem 5.1. // it satisfies (5.1), then
(5.2) 7reJ^([Z, Y],;[£\ «<],) V0, 0 < 0 < 1.
Proo/. Let 0 be fixed. Set s = 1/(20); then [X, Y]d is the space
described by u (0) as u describes W ( — oo, +co; s] X, Y) = W. Therefore,
if a g [X, Y]e, there exists a u gW such that u(0) = a. Define v by
v(0 = 7t(u(t)) a.e.
Since tzgJ^X; X) (resp. J^(Y; «0, we see that
^GL2(Rt;^)(resp. |t|*v(t) = ti(\t\s u(t)) g L2(RZ] &))
and that, with tx (resp. f}) denoting the norm of it in S£ (X, SC) (resp.
se(Y\ <&)\.
HvllL2(Re;ar) ^ * H«llL2(Re;jo,
H|T|^|lL2(RT^)^iS|||T|^||L2(RT.y).
Therefore v e W(-oo, + oo; s; #", «0 = tT and
(5.3) llwll^^max^.^lli*!^.
Then v (0) e [#*, &]e; since v -*• v (0) is a continuous mapping of W -»
-*- [8£t&]e and since v(0) = na, we have
(5.4)
II ^ ^ IIcar,^]© = c ll^ll"^'
which, together with (5.3), yields
||n ^||[^,^]0 ^ c max(#, /?) inf ||^H^, «(0) = a,
from which (5.2) follows. □
5.2 Interpolation of a Family of Operators
Theorem 5.2. Let it satisfy (5.1). Let nQ be a family of operators
(0 ^ Q ^ £0) satisfying
(5.5) 7r0eJ^(Z;^)nJ^(y;^)

28 6. Reiteration Properties and Duality of the Spaces [X, Y]$
(and therefore also satisfying (5.2)). Assume that
(5.6) nQ a -* n a in 9C (resp. <&) "iaeX (resp. Y), as q -» 0.
Then
(5.7) 7iQa->7ia in [&, W]eVa e[X, Y]e, as q-> 0.
Proof. We may assume it = 0.
With the notation of the proof of Theorem 5.1, we have:
(5.8) \\7t0a\\isCt9h ^ (constant) (\\nQu\\L2(Rt.X) + \\tiq(\t\su) ||L2(Rt.y)).
We shall verify that
(5-9) II^«IIlkh,;Z)-*0.
In the same manner, we would verify that ||7re(|r|s u) ||L2(Rt;y) -*• 0
and therefore (5.8) yields the desired result.
But (5.9) is a consequence of Lebesgue's Theorem; indeed, according
to (5.6) (with n = 0), Ttg u(i) -*> 0 a.e. in 9C, and still according to (5.6):
NJ*(*;*) = constant, therefore \\nQu(t)\\x ^ c \\u[t)\\x. D
Remark 5.1. Because of the preceding properties, the spaces [X, Y]e
are also called interpolation spaces between X and Y (but there are other
intermediate and interpolation spaces — in particular wow-Hilbert
spaces). D
6. Reiteration Properties and Duality of the Spaces [X9 Y]e
6.1 Reiteration
Let 0O and Bx be fixed in ]0, 1[, with
(6.1) 0o<Oi.
Then
(6.2) [X;Y]eo<=[X,Y]ei.
Proposition 6.1. The space [X, Y]eo is dense in [X, Y]6l.
Proof. By diagonalization (see Proof of Theorem 2.3), we see that °U
is an isomorphism of [X, Y]e onto §1_e = {v \ v e t), A1"0 v e t)} and
^x_eo is dense in f^.^. □
Therefore, we may apply the theory of intermediate — or
interpolation — spaces to the couple {[X, Y]eo, [X, Y]6l}.
Theorem 6.1. For all 6 e]0, 1[, we have
(6.3) [[X, Y]fl0, [X, Y],,], = [X, Y]u_e)eo+eei,
with equivalent norms.

6.2 Duality 29
Proof. We use the diagonalization °U. Then, the property to be
demonstrated is equivalent to:
(6-4) [9i-0o> Pi-eje = 9i-«i-0)0O+00i)'
But l)1_0o is the domain m ^ space iji-6l of the operator v -» A0l~e° fl,
therefore (by definition of the spaces [X, Y]e) the space [I)i_0o, Iji-aja
is the domain in the space iji-01 of the operator v -» ^(i-e)(°i-0o) V an(j
therefore coincides with the v's such that
A1"01 (A*1-^1-*0' w) e I), whence (6.4). D
Property (6.3) is called reiteration property or stability property: by
successive applications of the "operation" {X, Y} -» [X, Y]0, for
various values of d, we recover a space of the same type (for different
values of the "interpolation parameter" 6). □
6.2 Duality
Since X c [X, Y]0 c Y, each space being dense in the following
one, we have, by duality (without any identification between the space
and its dual):
Y' c [X, Y]'e c X'9
each space being dense in the following ones. We have the following
duality theorem:
Theorem 6.2. For all 0 e]0, 1[,
(6.5) lX,Y]'9 = [Y'tX']x-9t
with equivalent norms.
Proof. Set (still in the notation of the diagonalization of the proof
of Theorem 2.3):
i)s = {v\lsvefy, seR
of arbitrary sign (and the norm of the graph ||Asfl||ij).
For the demonstration, we identify — which is permissible —
Y' with Y, therefore i)' with i) = i)0. Since °U is an isomorphism of X'
onto !)_! (dual of i^) and of [X, Y]'e onto i)e_l (dual of ^!_e), the
property (6.5) to be demonstrated is equivalent to
Vi = ft* *)-i]i-e>
which is a consequence of the definition of the spaces [X, Y]e, since i)
may be defined as the domain of v -» X v in i)_x. D

30 7. The Spaces Hs(Rn) and Hs(r)
1. The Spaces Hs(Rn) and Hs(r)
7.1 ff(RM)-Spaces
First, we apply the concepts of Section 2.1, with
X = Hm (Rn), integer m > 0, Y = H° (Rn) = L2 (Rn).
According to Theorem 1.2, IF defines an isomorphism of X onto
£ = tim{Rm) ={w|(l + \y\2)m/2veL2(Rny)}
and of Y onto L2(R") = Y. Then, if /I is the operator associated to
the couple {X, Y} as A is associated to the couple {X, Y} (see
Section 2.1), we have:
Av = (1 + \y\2)m/2v,
from which we immediately obtain
(A)ev = (1 + |y|2)a,">2v.
We define the space (s e R)
(7.1) Hs(Rn) = {v\veST'(Rn),(l + \y\2)s/2 v eL2(Rj)}((1))
and provide it with the norm
(7.2) l|w||HW) = ll(i + |y|2)s/2^llL2(R;).
which makes it a Hilbert space.
Remark 7.1. We insist on the fact that s may be negative in
definition (7.1). We have:
(7.3) Hs(Rn) c#0(Rn) c#a(Rn) if a < 0 < s.
The spaces i/s(RM) are often called "fractional order Sobolev spaces". □
We may now state
Theorem 7.1. In the notation (2.7) of Definition 2.1, we have
(7.4) [Hm (Rn), H° (R% = #(1"e)m (RM),
with equivalent norms.
Remark 7.2. In general, letL2 (d/u) be the space of (classes of) square
integrable functions on a locally compact space for the measure d[x\
((1>> Equivalently, if*(Rn) may be defined as the space of Fourier antitransforms
of the measurable functions v such that (1 -\- |y|2)'/2 v G L2 (RJ).

7.1 tfs(Rn)-Spaces
31
if M is a — say, continuous — positive function, let
L2M (dp) = {/ | / measurable, M f e L2 (dp)}.
Then the dual space (L2M (dp))' is identified with L2lfM) (dp). Consequently,
(from Definition 7.2) we have:
Theorem 7.2. The space H° (Rn) being identified with its dual (or
antidual), we have, for all s > 0:
(7.5) (Hs(Rn))' = H-*(Rn). □
Remark 7.3. This is also a consequence of the duality Theorem 6.2. □
Local properties of Hs (Rn)-spaces.
Theorem 7.3. For all s e R and all cp e@(Rn),
(7.6) u -> cp u
is a continuous linear mapping of Hs(Rn) into itself.
Proof. 1) If the property holds for Hs(Rn), it holds (by transposition
and with (7.5)) for H~s(Rn) as well. Therefore, it is sufficient to prove
the result for s > 0.
2) Choose integer m ^ s. It is easy to verify that (7.6) is a
continuous linear mapping of Hm (Rn) into itself and of H° (Rn) into itself.
Therefore, by interpolation (Theorem 5.1) it is a continuous mapping
of [Hm (Rn), H° (Rn)]d into itself. It is permissible to choose 6 such that
(1 — 6) m = s. If we do so and use (7.4), we obtain the desired result. □
Remark 1 A. The space @(Rn) is dense in Hs(Rn), Vs.
Indeed, X is dense in [X, Y]e and therefore if X0 a X, X0 dense
in X, we have: X0 dense in [X, Y]e. It is sufficient, for s ^ 0, to apply
this with X = Hm(Rn) (integer tn^s), X0 = @(Rn), Y = L2(Rn);
for s < 0, we may reason by duality or take X = L2 (Rn) and Y = H~m (Rn).
It is also possible to verify that the usual approximation by regularization
and truncation is valid in Hs(Rn). 0
Theorem 7.3 can be supplemented with
Lemma 7.1. Let s be real and let cp be given in @(Rn). Then there
exists a constant c and a constant cSt<p which depends on s and cp such
that, yueHs(Rn):
\\<PU\\HHRr>) ^ C (SUP | (p |) || U
We shall give two proofs of this lemma, one now and another in
Remark 10.6.
Proof. Thanks to Remark 7.4 we may assume that we^(R").

32 7. The Spaces Hs(Rn) and Hs{r)
/s
Since <pu = y*u and since the norm of v in Hs (Rn) is equivalent
to ||(1 + |f|)s£||L2(Rn), we have
IIV«lln.(R-) = c|(l + \S\YJ4>(fl)*(S-ri)dril ^
II Rn ||L2(R«)
^ ||J 0fo)O + |f-1?l)S«(f-1?)^| +
||Rn ||l2(R")
+ c I / #fo) ((1 + \Z\Y - (1 + If - >?l)s) * (f - if) <fy[
||r" ||l*(R")
But
II J>fo)(l + lf-,l)'d(f-,)l*J
IIR" ||l.2(R")
= I j #(f -0 gW <»| (where g(*) = (1 + \t\)su(t)))
|r» ||l2(rh)
= \\<P*g\\mR») = (using Parseval's equality)
= ll9>£llL2(Rn)^ (sup|9)|)||^||L2(Rn)^
^ c(sup 1991) ||«||h.(r»).
The inequality of the lemma is therefore a consequence of the
following inequality:
/ 4>(v) (0 + If l)s - (i + If - v\Y) «(f - 7) <fy
< c |S| / f |9| (l + |,|)i«-ii |0fo)| <fy\ |«||H..i(R.).
\R"
which in turn follows from the elementary inequality:
Ki + ifi)«-(i + |{-,i)«i^
?*\s\\ri\(l + \£-V\y-l(l + (1 + |^|)I-H). D
We also note that from Theorem 7.1 and the definition of Hs(Rn)
it follows that:
Lemma 7.2. Let 99 he given in 2>(Rn) and let D" he a derivative of
arbitrary order'\ot\. Then
u -» <pD*u
is a continuous linear mapping of Hs(Rn) -» HS~^(R"), for all real s.
Remark 7.5. According to the reiteration theorem, if sx and s2 e R,
[HSi(Rn),HS2(Rn)]d = H(l-6)Si+6S2(Rn).

7.2 Traces on the Boundary of a Half-Space 33
7.2 Traces on the Boundary of a Half-Space
Consider now Hm(Q) for Q = {% \ xn > 0}. We note that:
( if we^^ooj^tRr1))
(7.7)
dmu
and -—eL^O.oo;/^:1)),
dx„
then —-eL2(0, oo; i/m"-/(Rr1)).
Indeed, if we apply the intermediate derivatives theorem
(Theorem 2.3), with a = 0, b = + oo,*„ = t,X = tf^R^r1), Y = i/()(R71),
then
OX„
from which (7.7) follows, since according to (7.4) (replacing n by (n — 1)):
Thus, from Theorem 1.3, we obtain
Theorem 7.4. For Q = {x | *„ > 0}, *Ae s^wce #m(£) <fe/waJ fry (1.3)
or by (1.21) may also be defined by
(7-8) f fimu i
ff"(fi) = \u\ueL\Q}^]Hm{Rx71))t — el2(0, oo; tf0^:1))
(with equivalent norms). □
Therefore, still with X = Hm(Rnx71), Y = ff0^"1), the space
Hm(Q) may be identified with W{0, oo) (see Definitions (2.9), (2.10)).
Theorem 3.2 joined with Theorem 2.1 then yields the important
consequence :
Theorem 7.5. Let Q = {x \ xn > 0}. The space ^([0, oo[; Hm(Rnx71))
is dense in Hm (Q). The mapping
[dju l — 1
(7.9) u-+ -j«0)
of ^([0, oo [; i/m(R^"1)) -»(#m(R£r1))m e*te^s fry continuity to a contin-
m-1
uous linear mapping of Hm(Q) -» f| Hm~J~1/2(Rnx71), which is sur-
jective. J=0
Proof. It is sufficient to write out
\X, Y](j+l/2)/m — [Hm(Rx' )>H (Rx' )](j+l/2)/m
explicitly, using (7.4). D

34
7. The Spaces Hs(Rn) and Hs(r)
Our immediate aim is to extend, in appropriate fashion, the results
of this theorem to spaces Hm(Q) with Q an arbitrary open set in RM.
In fact, we shall impose very strong hypotheses on Q (and which are
amenable to numerous generalizations).
7.3 Hs (/>Spaces
For the remainder of this book, Q shall denote an open set in RM,
with
[ the boundary r of Q is a (n — 1) dimensional infinitely
n in\ J differentiable variety, Q being locally on one side of r (i.e.
| we consides D a variety with boundary of class C °°, the boundary
| being T).
We shall denote by dT or dx the surface measure on r, induced
by dx.
In general, we shall assume that
(7.11) Q is bounded
(except in particular cases such as Q = Rn or Q = half-space).
It is hopeless to try to extend Theorem 7.5 to the case of an open
set Q with (7.10) and (7.11) unless Hs(r) is properly defined. D
Definition of Hs(r). Let 0Jt j = 1, . .., v be a family of open
bounded sets in RM, covering r, such that, for each /, there exists an
infinitely differentiable mapping
*->Vj{x) =V
of Oj -> 2, = {y | y = {y't yn}, \y'\ < 1, -1 < yn < 1} such that q>j
has an inverse,
y-xplHy) = *
which is also an infinitely differentiable mapping of 2, -> (9Jt q>j mapping
<9jnQ->2+ ={y\ye£,yn>0}
(resp. (9jnCn->2_={y\ye2, yn<0}, resp. ^nr->^n{y„ = 0}).
Furthermore, let the following compatibility conditions hold: if
Oj n 0t 4= 0, there exists an infinitely differentiable homeomorphism
Ju of (pi{@i n Oj) onto q>j{(9t n Oj), with positive jacobian, such that
?A*) = Jtj(9t(*)) Vxe&tn&j.

7. 3Hs{r)- Spaces 35
Let {ocj} be a partition of unity on r having the properties:
otj e @(r) = space of infinitely differentiable functions on J7,
V
otj with compact support in Oj n jT, £ otj = 1 on r.
If « is a function on r (for instance integrable), we decompose
(7.12)
(7.13). « = !(*,«).
and define
(7.14) <p* fa u) <?, 0) = («, u) (<pjl{y'> 0)), / e J n {y, = 0}.
Since #, has compact support in r c\ 6j} the function 99* (#/ w) has
compact support in 2L n {yn = 0} and therefore we may also consider
99* (#, u) to be defined in R",~* by extending it by zero out of 2, n {yn = 0}.
u -► 99* (^ «)
is a continuous linear mapping of L1 (jT) -► L1 (R"r*), of 0 (jT) -► 0 (Ry~ *)
and extends by continuity to a continuous linear mapping of & (T1) -►
-►^'(R",-1) (easily verified by duality).
Now, we define
(7.15) H*(D = {u I tffa u) e ^(R^1) / = 1,. .., „},
which is valid for any real s.
Thanks to the local character of Hs{YFy7l) (see Theorem 7.3) we
see that the (algebraic) definition (7.15) is independent of the choice of
the system of local maps {&j} <pj} and of the partition of unity {<Xj}. □
Next, we may take the norm
(7-16) ||u\\Hs(n = (£||v* fa u) \\liKn7l\ ' ,
which, of course, depends on the system {(9J} q>j} otj}. We easily verify
that, with (7.16), Hs(r) is a Hilbert space and that the different norms
(7.16) are equivalent. We note that
(7.17) 2{T) is dense in Hs{r), s ^ 0.
Indeed, <p*(otju) = lim y)Jn, xpjne 2{R!l7l) and we may suppose
n-* oo '
that y)jn has compact support in | y' \ < 1, the limit taking place in
Hs{Kny7l) (see Remark 7.4; we arrive at compact supports in \y'\ < 1
by infinitely differentiable truncation), from which results that otj u
= limit in Hs(r) of ((pi)-1 (y)jn) e9{r). □

36 7. The Spaces Hs{Rn) and Hs{r)
Theorem 7.6. Identifying H°(r) with its dual, we have
(7.18) (Hs(r))' = H-(r).
Proof. It is sufficient to consider s > 0.
According to definitions (7.15), (7.16) and the Hahn-Banach theorem,
every continuous linear form u -» L (u) on Hs (J1) may be written, in
non-unique fashion,
(7.19)
j = i
fjeiH'MT1))' = H-°(Rny71).
If yje^R","1), has compact support in \y'\ < 1 and is equal to
one in a neighborhood of the support of (p*((XjU), we may write (7.19)
in the form
(7.20) L(«) = £<fjfj,ri (*,«)>•
J = l
Therefore,
(7.21) L(u)=(i*j,<p](y>Jfj),u\
and L may be represented by
V
Y^jt(Pj{Wjfj) =/•
j = i
But then y* <xkf e H-s(Rny7l), so that feH~s(r) by definition.
Reciprocally, if / e H~s(r), we define </, «> for w e@{T) (then for
w e Hs(r)) by £ <<%,• f,ocku} (for <%,••#* 4= 0) and we define <<%,• f,otku)
J.k
by returning, via local maps, to the duality between
H-s{Rny7l) and ^(Rjr1). □
We also have
Theorem 7.7. For all slt s2 6 R, sx > 52:
(7.22) [HSl(r),HS2(D]d = H(l-d)Sl+dS2(r),
with equivalent norms.
Proof. 1) Let ax > a2t f e]0, 1[. We show that
(7.23) [H* (D, Ha> (r)]f c #(1 "^)ai +a2* (D
(which yields the inclusion of the first space in the second in (7.22)).
Indeed, V/, w -*• (p*(ocju) is a continuous linear mapping of

7.3 H5(r)-Spaces 37
(by definition), therefore, according to the interpolation Theorem 5.1
and Remark 7.5, of
tHai(r),H*2(r)]e -> [Hai(*Z71),H*2(R;7% = H"-**1***2^1).
Therefore, if
us[Ha>{r),Htt>{r)\,
we have
and thus
we#(W)fll+*fl2(r), whence (7.23).
2) Since 2{r)a [Hai(r), Ha2(r)]it this last space is dense in
#(W)fll+*fl2(r), which together with Theorem 7.6 yields
#-(W)fli_*fl2(r) c [Hai(r),Ha2(r)]'z.
But according to the duality Theorem 7.2, this means
(7.24) #-(W)*i-**2(r) c [/Tfl2(r),/Tfll(r)]w.
Choose — a1 = s2, — a2 = slt 1 — £ = 0; then (7.24) yields the inclusion
from right to left in (7.22). D
Remark 7.6. Use of the Laplace-Beltrami operator.
It is possible to give a more intrinsic definition of the spaces Hs(r)
by using the Laplace-Beltrami operator Ar on r (see de Rham [1]).
Indeed, it will follow from Chapter 2, Section 3 that
(7.25) H2m (r)={u\ue L2 (71), A? u e L2 (F)},
any one of the norms defined by (7.16) (with s = 2m) being equivalent
to ||«||l2(t) + MmwllL2(r)- We may then define Hs(r) by interpolation.
Therefore, we also have (using (7.22) and (7.23)):
(7.26) Hs(r) = domain of (-Ar)s.
We use the eigenfunctions wi of — Ar on F (see for example Bere-
zanksi [4], Minakshisundaran-Plejel [1]):
(7.27) -Arwj = XjwJ> /= 1,...,
where the w/s are orthonormalized in H°(r).
If u is a distribution on r (u e^(.T)), let (u, Wj} be its Fourier
coefficients (relative to {wj}). Then, we obtain (s e R):
(7.28) H°(D = L\ueS,(D,(yX? |<«, ®,>|2)
1/2
< 00
D

38 8. Trace Theorem in Hm(Q)
8. Trace Theorem in Hm{Q)
8.1 Extension and Density Theorems
Theorem 8.1. Let Q satisfy (7.10) and (7.11). Then, for any integer
m > 0, there exists an operator p with the properties
(8.1) pe^(Hk(Q),Hk(Rn)) Vft, 0 ^ k ^ m,
(8.2) pu = u a.e. on Q VueH°(Q).
Proof. 1) Let /}0,/}lt . . .,/}v be a partition of unity in Q with:
p0 e2{Q), #, e@(D), .1 g j g v, ^ with support in 0, n £,
££,= 1 in Q.
j = o
Then for arbitrary u e H° (Q) we have:
V
U = /50U + Y,PjU-
We shall define p(Pju), from which we will obtain that p defined by
V
(8.3) pu = p0u + Y,P(Pju) (Po u extended by 0)
has properties (8.1) and (8.2).
2) Let ueHk(Q). The function qfiffiju) (see Definition of Hs(r)
in Section 7, for the introduction of cpj and 99*) is in Hk(Rn+)
(R+ = [y I y„ > 0}), with support in i?+ and vanishes in the
neighborhood of the boundary of i?+ distinct from {yn = 0}. We extend
99* (Pj u) with the help of the extension by reflection introduced in the
proof of Theorem 2.2 for Hm(Rn+).
Here we set:
[ v(x) if xn > 0
(8.4) p * v = I m
5>kv(*', -**„) if xn <0
(where #fc is defined by (2.23)).
Then p* <p*(f}jU)e Hk(Rn) if u e Hk(Q) for k ^ m and has compact
support in J. We may set:
(8.5) P(§Ju) = (<p*])-l{P*<p*](p]u)),
from which the theorem follows. □
Remark 8.1. The restriction to i2
(8.6) u -* rQu = restriction of u to Q,

8.2 Trace Theorem
39
is a continuous linear mapping of Hk(R") -► Hk(Q), for every integer
k ^ 0. D
Theorem 8.2. If Q satisfies (7.10) and (7.11), the space @(D) is dense
in Hm(Q).
Proof. There are two methods:
(i) approach (p*(^ju) (notation of the proof of Theorem 8.1) with
functions of @(M+) vanishing in the neighborhood of the boundary
of J2+ distinct from [yn = 0} (which is permissible according to
Theorem 2.1);
(ii) approach pu with 06e@(Rn) in the sense of Hm(Rn) (Remarkl.3),
then
u = rQ(pu) = limr^ 0Q in Hm(Q) (Remark 8.1)
and since rQ &e e2(Q), the result follows. □
8.2 Trace Theorem
The following result plays a fundamental role in the rest of the book:
Theorem 8.3. Assume that Q satisfies (7.10) and (7.11). The mapping
dJu
(8.7)
dvJ
dJu
j = 0, 1, . . ., m — 1
where ^—- = normal /-order derivative on J1, oriented toward the
\ dvJ \
interior of Q, to fix the ideas11 of Q}{D) -> (@(r))m extends by continuity
to a continuous linear mapping, still denoted
\dJu
of
dvJ
j = 0, . . ., m — 1
j^(0) Jnff"-j-i/2(r).
i = o
This mapping is surjective and there exists a continuous linear right
inverse
such that
(8.8)
g = {gj}^@g of f\H"<-J-Ui(r)^H">(Q),
j=o
dJ
or
1 In the following chapters, we shall often use the notation

40
9. The Spaces HS(Q), Real 5 ^ 0
Proof. 1) Let ue2{D). With the help of local maps, choosing, as
it is permissible to do, the local coordinates so that the image of (ocj u)
under q)j be
d #
we see that according to Theorem 7.5 and the definition of Hs(r) the
mapping (8.7) is continuous for the topologies indicated in the statement
of the theorem. According to Theorem 8.2, we may extend to Hm (Q).
2) The surjectivity follows from the surjectivity in Theorem 7.5.
The existence of the right inverse follows from Remark 3.3 (which gives
an explicit right inverse) or from the following general remark (but
particular to Hilbert spaces, which is not the case with the first method):
if X and X are two Hilbert spaces and if T e S£ (X; X), where #~
is surjective, then 3~ is an isomorphism of X^x? onto X (Ker^~
= kernel of F); but X being a Hilbert space, XIKqt*t is identified
to a subspace of X, say X0, and it is sufficient to "right invert" F
by the inverse of the isomorphism of XQ onto X, defined by ^. □
9. The Spaces HS(Q), Real s^O
9.1 Definition by Interpolation
By definition, we set
(9-1)
HS(Q) = [Hm(Q),H°(Q)]e, (1 - 6) m = s, m integer, 0 < 6 < 1.
Clearly, there are several items to be verified:
(i) up to an equivalence in norms, definition (9.1) depends only on s
and not on the choice of m (as long as (1 — 6) m = s);
(ii) when s is an integer, definition (9.1) gives back definition (1.1)
(still up to an equivalence in norms).
Such is the case when Q has a sufficiently regular boundary, as the
following theorems show.
Theorem 9.1. Assume that Q satisfies (7.10) and (7.11). Then HS(Q)
coincides (algebraically) with the space of restrictions to Q of the elements
of Hs(Rn).
Proof. 1) Consider the operator p of Theorem 8.1. We have:
pe^(H° (Q); H° (Rn)) n £(Hm (Q); Hm (Rn)),
therefore, by the interpolation Theorem 5.1:
p e -S?([ff" (Q), H° (Q)]e; [Hm (R"), H° (R")]e).

9.2 Trace Theorem in HS(Q) 41
But according to definition (9.1) and Theorem 7.1, we have:
pe^(Hs{Q)]Hs(Rn)).
2) In the same way, using the "restriction to Q" operator rQ} we
have:
rQe^(Hs(Rn);Hs{Q)).
3) Then every u e [Hm(Q), H° (Q)]e = HS(Q) is of the form:
u = rQpu,
i.e. is a restriction to Q of p u e Hs(Rn) and every u = rQv} v e HS(R"),
belongs to HS(Q). D
Theorem 9.2. The norm of HS(Q) defined by (9.1) is equivalent to the
norm
(9.2) |H|n.(0) = Inf || U\\HS(Rn)> U = u a.e. on Q.
Proof. The norm (9.2) amounts to providing HS(Q) with the norm
of the quotient of Hs (Rn) by the closed subspace of functions vanishing
a.e. outside of Q. Then, HS(Q) is a Hilbert space for the norm || u\\H,m
associated to (9.1) and for (9.2); and since
IMIh.CO) ^ ll^^llH-Ot-) ^ C \\U\\hHQ)>
we have the equivalence of norms. □
Also, note
Theorem 9.3. Assume that Q satisfies (7.10) and (7.11). Then @(Q) is
dense in HS(Q), s ^ 0.
Proof. According to Remark 2.6, Hm{Q) is dense in HS(Q), ((1 —Q)m
= s). Hence, the theorem follows as a consequence of Theorem 8.2. D
Furthermore, we deduce from the preceding results that
Hs{Q) = [Hk+1(Q),Hk(Q)]l_e if s = k + d, O<0<1,
integer k ^ 0,
which allows us to define HS(Q) by interpolation "between two
consecutive integers". D
9.2 Trace Theorem in HS{Q)
Theorem 9.4. Assume that Q satisfies (7.10) and (7.11). Then, the
mapping
f dJu ,
1 =0,1, ...,//
(9.3)
a^

42
9. The Spaces HS{Q), Real s ^ 0
o/ 2 (D) -*■ (2 (.T))m extends by continuity to a continuous linear mapping
8Ju
dvJ
j = 0,...,ft\ of Hs(^)->nffW_1/2(^).
where fi is the greatest integer such that
(9.4) ii < s - \.
77^ mapping u
dvJ
j = 0, . . ., ju,\ is surjective and there exists
a continuous right inverse mapping of
f[Hs-J-1!2(D-+Hs(Q).
j=o
Proof. 1) Let ueHs(Q). The space HS(Q) has a local character
(as in Theorem 7.3), i.e., if j}0,(} lt..., fa is the partition of unity
introduced in part 1) of the proof of Theorem 8.1, then
faueHs(Q), 0^/^ v.
We may reason on U e Hs(Rn), U = u a.e. on Q, as well (according
to Theorems 9.1 and 9.2). We may assume that fa e @(Rn), with support
V
in Oj, 1 ^ / ^ ?,£/?/= 1 in a neighbourhood of D. Then
j=o
faU eHs(Rn), fa U has support in 0j, 1 £ j £ v (p0U with compact
support in Q).
u-xpKPjU)
defined by q>j (local map), is a continuous linear mapping of
H>»(R:) (resp.^0(R:))->^(Rj) (resp. H° (Rj))
and therefore, by interpolation, of Hs(Rnx) -> HS(R$). So that:
(9.5) q>HPjV)eH'fflVi-
2) But it is easily verified that we may "separate the variables" in
the definition of Hs(Rn) (compare with (7.2));
(9.6) H*W)-{v\veL2(R^\H'&7%\yn\^
where
f v(x\ yn) = Fourier transform in xn of v (xf, xn), therefore equal to
(9.7)
v:
f c
-f
In J
exp(-ixHyH) v(x', xn) dxn.

9.3 Interpolation of Hs(fi)-Spaces 43
Indeed, applying the Fourier transform in x' to definition (9.6),
we obtain
(1 + \y'\2y'2veL2(R;),
\ytt\°veL2(R;),
which is equivalent to (1 + |y|2)s/2 v eL2(RJ).
3) We apply Theorem 4.2 with t = xn (r = y„) and
X = HS(R^71), Y = H0(R"X~1)-
We obtain that
5*
3y*
itiWj U)) k-oefl-1-1'^:1), s - k - i > 0,
from which by " getting onto r again'' (and choosing the local coordinates
d d
so that -7— becomes
dv dyn
~d7
J-n = 0,
(PjU)U-^;iPj»)\eH'-'-u*(r).
4) The surjectivity results, by the same procedure, from the sur-
jectivity in Theorem 4.2. The continuity of the right inverse results,
either from an explicit construction starting with (4.8) or from the
general remark at the end of the proof of Theorem 8.3. □
Condition (9.4) cannot be weakened:
Theorem 9.5. Assume that Q satisfies (7.10) and (7.11). Suppose
(9.8) ii ^ s - \.
Then, for any <p e@(r), <p # 0, the linear form
f d^u
(9.9) u->\ <pdr, ue9{D)
J dv"
r
is not continuous for the topology induced by HS(Q).
Proof. As above, we are brought back, via local maps, to the
application of Theorem 4.3. □
9.3 Interpolation of Hs (fi)-Spaces
Theorem 9.6. Assume that Q satisfies (7.10) and (7.11). Then
(9.10) [HSl(Q),HS2(Q)]6 = ff(1-6)Sl+*2(fi)
for all st > 0, s2 < slt 0 < 0 < 1 (with equivalent norms).

44 9. The Spaces HS{Q), Real 5 > 0
Proof. Let integer m ^ max (slts2). Then
#Si(£) = [ff"(0),ff°(0)]e<> (1 -Ot)tn = st, i = 1,2,
so that
[ff " (0), H°> (Q)]e = IIH" (Q), H° (Q)]6l, [ff» (Q), H° (fi)]J,
= [#"■(£), #°(£)](1-,,)(,1+(M,2
(the last step follows from Theorem 6.1), from which (9.10) follows. □
Remark 9.1. From (9.10) and (2.43), we deduce the interpolation
inequality
IMItf<i-a>.i + «i(fi) ^ C(0,Sl9S2) Hltf^) ||«|lSr-2(fi)- D
Theorem 9.7. Assume that Q satisfies (7.10) awrf (7.11). 77&ew, *Ae two
conditions
(9.11) uEHs(Q),
(9.12) *'/ m is «» wfeger ^s,ue Hs~m (Q) tDpueHs~m (Q), V\p\ = m,
are equivalent.
Furthermore, the norms
II«IIh.cd> and (IMlJ-«a»+E ll^«llS.-«ai))i1/2
are equivalent.
Proof. 1) It is sufficient to show the result for m = 1 and to apply
it iteratively.
2) Let Wj and m2 be integers, mx > s, m2 ^ 5, m2 ^ 1.
Since 3/3 #t is a continuous linear operator of HmJ(Q) -» Hmj~1(Q)t
j = 1,2, it is, by interpolation, a continuous operator of
[#mi (Q), ff ma (fi)]e -> [Hmi -'(Q), Hm2"* (Q)]9.
Therefore, taking (1 — 6) mt + 0 m2 = s, we see that
-£-eJ?((H'(Q);H'-i(Q)),
which shows that (9.11) implies (9.12).
3) Conversely, let
ueH*-l(Q) with —eH'-^Q), i=\,...,n.

9.4 Regularity Properties of ifs(£)-Funktions
45
According to the proof of Theorem 9.1 and the construction of p
(see also Theorem 8.1), we can find an operator p having the properties
(9.13)
(9.14)
(9.15)
Then:
Pe&(Ha{Q)\Ha{W))> O^a^s,
pu = u a.e. on £, yiueH°{Q),
dxt \dxj
where q e ^(Ha(Rn); Ha(Rn)), 0 ^ a ^ s - 1
qu = u a.e. on Q Vwe H° (Q).
pueH'-l(Rn),
d [ du\
3«i \dxtJ
which implies pueHs(Rn) (immediate verification by the use of the
Fourier transform), from which the theorem follows. D
9.4 Regularity Properties of Z/5(f?)-Fuiictioiis
It is natural to compare HS(Q)-spaces to other spaces, the simplest
being LP(Q)-spaces, p + 2, and the usual spaces of differentiable
functions on D.
The comparison of HS(Q) with LP(Q) (or with Sobolev-spaces
constructed on LP{Q)) results, for integer s, from the Sobolev inequalities,
for which we refer the reader to Sobolev [2] and Gagliardo [1].
For non-integer s, there are, in this connection, the "fractional
Sobolev inequalities"; we refer the reader to J. Peetre [12] and to the
bibliography of this work.
Here, we give a very simple result pertaining to the comparison
between HS(Q) and
(9.16) C°(D) ={v\v continuous on D},
which is a Banach space for the norm max | v (x) \.
Theorem 9.8. Assume that Q satisfies {lAG)-and (7.11). Then, if
(9.17)
we have
(9.18)
with continuous injection.
s>- (£cR"),
H°(Q) cC°(D)

46 9. The Spaces HS{Q), Real s ^ 0
Proof. Let ueHs(Q). We introduce (see proof of Theorem 9.1)
f v = pueHs(Rn)
(9.19)
[ v = u a.e. on Q.
Let v be the Fourier transform of v. Then
^(l + lfD-'U + lfD'OeiMK-)
and
I|v||LI(R;) ^ C |M|ff.(R»).
Therefore v = (inverse) Fourier transform of v eL1 (Rfj is (a.e. equal
to) a continuous function (vanishing at infinity), so that by restriction
to Q we obtain the desired result. D
Remark 9.2. The result of Theorem 9.8 cannot be improved. For
example, if n = 2, s = 1, H1 (Q) is not contained in C°(D)\ indeed,
the function
*-> log |log |*||,
defined on the disk Q:\x\<i, belongs to HX{Q) but not to C°(D). D
Remark 9.3. The injection (9.18) is to be understood in the sense:
Hue HS(Q), it is a.e. equal to an element of C°(D), identified to u. □
Remark 9.4. We have implicitly shown that
[ if (9.17) holds, then Hs(Rn) a space of continuous functions
(9.20) | vanishing at infinity (and even, more precisely, Hs (Rn) a ^L1
I = image of L1 (Rn) under Fourier transformation). 0
From Theorem 9.8, we immediately deduce the following corollaries:
Corollary 9.1. Assume that Q satisfies (7.10) and (7.11). Then, if
n
(9.21) s > h m, integer m > 0,
we have
(9.22) HS(Q) a Cm(D) = space of m-times continuously differentiable
functions on D.
(with continuous injection, Cm(D) having the norm £ max|D"v(x)\\.
Corollary 9.2. Assume that Q satisfies (7.10) and (7.11). Then
(9.23) f]Hk(Q) = 9(Q).
Remark 9.5. If X is a Hilbert space, we define (integer m ^ 0)
Hm{Q\X) ={v\DpveL2{Q\X)V\p\ ^ m},

10.1 Domains of Semi-Groups 47
where L2(Q;X) = space of (classes of) square integrable functions
on Q, taking their values in X; provided with the norm
( I \\\Dpv(x)\\2xdx) ,
it is a Hilbert space. Of course
H°(Q]X) =L2(Q]X).
Then, for real s > 0, we define
HS(Q;X) = [Hm(Q;X),H°(Q;X]e, (1 - 0) m = s;
it can be shown that this space depends only on s (and not on m), up
to equivalence of norms.
Further, it can be shown (by the same type of methods as in the
preceding proofs) that, if Y is a second Hilbert space (as in Section 2),
we have
(9.24) [HSl(Q] X)tHS2(Q) Y)]e = H(1-Q)Sl+dS2(Q', [X, Y]e).
In the same way, the scalar setting considered in the following
sections extends to the vector case, as in (9.24). D
10. Some Further Properties of the Spaces [X, Y]9
10.1 Domains of Semi-Groups
Let X, Y be two Hilbert spaces satisfying (2.1). Let G(t) be a
continuous semi-group on Y, that is:
f G(t) e£>(Y; Y) "it ^ 0, G(0) = identity = /,
(10.1) ] VyeY, t->G(t)y is a continuous mapping of t°^.0-+Y,
| G(t)G(s) = G(t + s) V2,s ^ 0.
Let A be the infinitesimal generator of G(t), with domain D(A)
(see Hille-Phillips [1], Yosida [2]), a Hilbert space for the norm of the
graph (||y||!+My|||)1/2.
We assume that:
(10.2) D(A) = X (with equivalent norms). □
Remark 10.1. There exists an infinity of semi-groups having property
(10.2); for example, A being defined by (2.5) and (2.6) (and there is
an infinity of such id's), we can take
(10.3) G{t) =exp(- tA)
(or G(t) = exp(itA), unitary group). D

48
10. Some Further Properties of the Spaces [X, Y]q
Remark 10.2. At the risk of having to change G(t) to e~<ot G(t), for
a suitable co (which changes A, but not D(A)), we may always assume
that || G(t) ||#(y>,y) is bounded. D
Theorem 10.1. Let X, Y satisfy (2.1). The three following statements
are equivalent (0 < 6 < 1):
(10.4) ae[X,Y]99
(10.5)
a = u(0), t*ueL2(0, oo;X),
du n 1
*• — el2(0, oo; Y), 0 = — +*,
at 2
(10.6) f-^GM a - a) eL2(0, oo; Y).
Furthermore, the norms
IMItx.n. <™<* (ll«llr + /<2<0-1)l|G(<)«-a||J
are equivalent.
^I2
dt
Proof. 1) (10.5) => (10.6).
Set
(10.7)
du
and note that t" f eL2(0, oo; Y).
Then (solution of Cauchy's problem; see for example Yosida [2])
u(t) = G{t) a + | G(t -a) f(a) da
o
and therefore
t
G(t) a - a = u(t) - «(0) - \ G(t - a) f(a) da,
o
so that
t t
\ r \ r
(10.8) r^Git) a - a) = — \u'(a) da \G(t- a) f(a) da.
o o
Therefore
\ri(G{t)a-a)\\Y-Z —
%>{a)\\Ydo+^[\\t{o)\\Ydc

10.1 Domains of Semi-Groups
and (10.6) will follow as a consequence of
Lemma 10.1. For <x < \\
49
(10.9)
t*\\\g(<y)da
^ (constant) \\t«g(t)\\L2i0tOD).
L2(0.oo)
Proof. Setting t = ex, a = ey, e(*-1/2)*g(e*) = g{x), (10.9) is
equivalent to
f g(«-l/2)(*-y)|Jy)^y
But
where
^ (constant) |||||L2(R).
L2(R)
J e(*-i/2)(X-y) g(y)dy = E*g{x),
£(*)-
[exp(# - i) #, # > 0,
0, *^0.
Since 6 < 1, we have # < £ and therefore
TOD
[E{x)dx = —L-,
J i-«
whence (10.9) (and since £(#) ^ 0, #»e fos/ constant in (10.9) is
!/(*-"))• 0
2) (10.6) => (10.5).
Let a satisfy (10.6); construct u as follows:
(10.10)
«(*) = q(t)
1 f
— G(a) a da,
q e C1 ([0, + oo [) and has compact support,
?(0) = 1
We have: «(0) = a. We have to show:
(10.11) f 4«eZ,2(0,oo; Y),
(10.12) f«'el2(0, oo ;Y).

50 10. Some Further Properties of the Spaces [X, Y]e
But in general (see Yosida [2])
so that
Alj G(a)ada\ = G(t) a - a,
t*Au = q{t)f-l(G{t)a - a)
and (10.1.1) follows from (10.6). Next
= f'T
G (a) a da I + q v' if
v{t) =-L|G(a)«
da
and (10.12) will obviously follow from
(10.13) fn'el2(0, oo;Y).
But
v'(t) =—G(t)a- —
G(a) a da
1 1 f
= — (G(t) a — a) \ (G (a) a — a) da
and therefore (10.13) will follow from
w
(t) = t*~2 j(G(a)a - a)daeL2(0,oo;Y).
Now G(a) a — a = a1'<x(p(a), cp e L2(0, oo; Y), therefore
w
(t) =t*~2 ja1-acp(a)da>
and (since a ^ t)
I "Wlh
^^yfcT-lly
»lly^*
from which the result follows, by (10.9). D
3) (io.4) => (10.6).
Since (10.5) o (10.6) and (10.5) is independent of the semi-group
G(t) (as long as D(A) = X), it is sufficient to show the equivalence
of (10.4) and (10.6) for one particular semi-group G(t); we consider
(10.3). Using spectral decomposition, (10.6) is equivalent to (setting

10.2 Application to HS(R")
51
b{X) = %a(X))
+ 00 +00
I j *-»>(! - e-'Y \\b(X)\\laydtdv(X) < oo.
0 A0
Since j t2(Cf~l) (1 — e~tX)2 dt = c A1"2*, the above integral becomes
o
+ 00 +00
c J A1"2* ||6(A)||Jtt)^(A) = ' J A2^-e> ||6(A)||2(A)^(A)
Ao
= C fl
[X.rie'
which gives the desired result and the equivalence of norms. □
Remark 10.3. The equivalence of (10.5) and (10.6) extends to the
case of Banach spaces and with L2 replaced by Lp. □
Remark 10.4. Let Gx, G2, ..., Gn be n continuous semi-groups on Y,
which are commutative, i.e.
(10.14)" Gt(s) Gj(t) = Gj(t) Gt(s) Vi.j.s.t.
Let A i be the infinitesimal generator of G, with domain D (At). Suppose
(10.15)
X = f]D(A()
i=l
I 2 £ 2\1/2
\\x is equivalent to I ||**||y + £ M***||y)
(10.16)
Then, if we assume \\Gt(t) \\&(y;y) ^ (constant) W, we have:
condition (10.5) is equivalent to
f-^Gtfla - a)EL2(0,oo; Y) V* = 1, . ..,n.
Indeed, (10.5) => (10.16) is obvious since (10.5) => (10.6). To show that
(10.16) => (10.5), we "right invert" a by the formula (compare with
(10.10)):
(10.17) u(t)=q(t)
Gx (a) da
G0 (a) da . . .
Gn{a) da • a..
We finish the proof as above. □
10.2 Application to H5(Rn)
We apply Theorem 10.1 and Remark 10.4 to Hs(Rn), for 0 < s < 1.
We have:
(10.18)
Hs{Rn) = [Hl (R»), H°(Rn)]e, 1 - 6 = s.

52 10. Some Further Properties of the Spaces [X, Y]e
We consider the translation semi-groups (in fact, groups):
(10.19) Gt(t) f(x) = f(xlt. . ., xi_ltxi + t, xi+l> ...,xn)
in
(10.20) Y = L2{Rn) =H°(Rn).
The theory applies. Therefore, according to (10.16), we have:
u e Hs{Rn)o f-^Gtit) u{x) - «(*)) e L2(0, oo; Y)
i.e.
00
f t*-"dt\ \u(xl9 .. .,xt + tt. . .,xn) - u(x)\2dx < oo.
0 R»
Therefore, since 2{(x — 1) = — (2 s + 1), we have:
Theorem 10.2. For 0 < 5 < 1, the following conditions are equivalent:
(10.21) ueHs(Rn),
ueL2 {Rn) and
00
(10.22) \ jt-<2s+1>dtj\u{xlt...txi + tt...txn)-u{x)\2dx<oo,
0 R»
i = 1, . . ., n.
Furthermore, the norms IMIhw) an^
MMIl2(R»)+ ^\t-^'Ht^\u{xu...txi^tt...txn)^u{x)\2dx\
are equivalent. D
Remark 10.5. Conditions (10.22) may be replaced by the equivalent
conditions:
f ueL2{Rn),
J J 7^
y\
n+2s
\u(x) — u(y)\2 dxdy < oo
RnXR"
the norm in Theorem 10.2 being equivalent to
llL2(R")
+
J J l*-y
\n+2s
\u(x) — u(y)\2 dxdy]
Ll/2\
RnxR"
This equivalence may be verified by Fourier transformation, for
example. □

10.2 Application to Hs(Rn) 53
Remark 10.6. We now give another (neater) proof of Lemma 7.1.
Let us define
(10.23) A0 = Hm+1 <zA1=Hm = B0czBl = Hm~\
where m is a positive or negative integer.
We assume that
(10.24) 5 = m + (1 -0).
Let G(t) be the semi-group defined in all these spaces by
G(J^) = exp(-*(l + |£|))0(0.
Then A0 (resp. B0) is the domain of the infinitesimal generator of G
considered in Ax (resp. Bx).
Consider two Hilbert spaces C0 cz C1 and let n be given to satisfy
(10.25) 7ce^(A0;C0) n^i;^)
and
(10.26) \\nu\\CtZ<x\\u\\At + p\\u\\Bt9 VueAt, * = 0,1.
Example. We let C0 = A0, Cx = Alt n = the operator u -> ipu,
(p given in 2(Q). With the choices (10.23), we have (10.26) with
(10.27) * = sup|p(*)|. D
The hypotheses (10.25) and the interpolation theorem imply that
n eJ?{Ae; Ce), where A0 = [A0,A1]e, etc....
In the example, Ae = Hs(Rn) (due to (10.24)) and Ce = A0t B0
The general result is that we can "interpolate" (10.26), namely
that there exists a constant y such that
(10.28) || 7tu\\Ce^ocy\\u\\Ae + j8y||«lk VueAe.
Lemma 7.1 follows from (10.28) applied to the example.
Proof of (10.28). Define (see (10.10))
v(t) = 9tu(t) = (— G(a)uda\q(
(10.29) v(t) =3lu(t) = (— G(a)uda]q(t).
Then, if we define WA (analogous definitions for WB and Wc) as
the space of functions u such that
du 1
f«eI2(0,ooMo), <"«' = t* — eL2(0,oD,A1), 6 =- + <x,
at 2t

54 11. Subspaces of HS(Q). The Spaces HftQ)
provided with the norm
= (II f -
llL2(0,oo,>lo)
+ \\r
IIL2(0, oo
U/2
.AiV »
we have
(10.30) 0le&{A9;WA) and 9teX[B9\WB)
(since A0 and B0 are defined in Ax and B1 through the same semi
group G). If we define
w(t) = n v (t),
then, using (10.26):
and since
Mlnrc ^ Ct(K\\v\\WA + C±P\\v\\Wb
^ c2oc\\u\\Ae + c2p \\u\\Be
7ZU\\C„ £C\\W\
WC
the result (10.28) follows. D
10.3 Application to J5P(0, oo)
We now apply Section 10.1 to
H*{0, oo) = [#'(0, oo),L2(0, oo)]„ 1 - 6 = s.
We consider the semi-group G(t) defined in L2(0, oo) by
G{t)f(x) = f{x + t) a.e., x > 0,
with infinitesimal generator A = , having domain H1^, oo).
dx
We obtain
for 0 < 5 < 1, the condition "u e Hs(0, oo)" is equivalent
to u e L2(0, oo) and
(10.31) {
\ t~<2s+1)dt j \u(x + t) - u(x)\2dx < oo. D
11. Subspaces of Hs(ii). The Spaces H'0(Q)
11.1 #S(fi)"Spaces
Since the mapping u
dJu
dvJ
0 < / < 5 — —I (see Theorem 9.4)
~ ' 2 v '
vanishes on 2{Q) and is a surjection of
0£j<s-l/2

11.1 H'0(Q) -Spaces 55
there results that, if s > ^, the space 2(Q) is not dense in HS(Q). In
general, we shall set:
(11.1) HS0(Q) = closure of &{Q) in H'{Q).
We have
Theorem 11.1. Assume that Q satisfies (7.10) and (7.1.1).
The space 2(Q) is dense in HS(Q) if and only if
(11.2) s^i
(then HS0(Q) = HS(Q)). If s > \t we have: H50(Q) is strictly contained
in HS(Q).
Proof. 1) There only remains to show that 2(Q) is dense in HS{Q)
if (11.2) is satisfied. Since (according to the general properties of the
spaces [X, Y]e):
Hl/2(Q) c HS(Q), Hl>2(Q) dense in HS(Q), if s < \%
it is sufficient to show that
(11.3) S(Q) is dense in Hl>2(Q).
2) For the time being, we admit
Lemma 11.1. Let X and Y be two Hilbert spaces satisfying (2.1).
Consider the space "T of v's such that
(11.4) veL2(Rt;X), |r|1/2 v eI2(Rt; Y)
+ oo
where v(r) = /— exp( —i tr) v(t) dt\, with the (hilbertian) norm
*w=tM
— 00
ii 2
(11-5) {\\v\\URtlXy+\\\t\il2v\\lHR,;r))il2.
The subspace of q> e Of (Rt; X), with (p = 0 in a (variable) neighborhood
of t = 0, is dense in the space ir. D
Now, we show (11.3). Via local maps, we are brought back to the
following: let
veH^2(Rn+)t
with compact support in R+; we need to show that v = lim^, <p5 e
e®(R\), limit in #1/2(Rn+). ^°°
Using an extension operator of Hs (R+) -» Hs (RM), which exists for
all s, we are led to:
f let v e Hl/2 (Rn); show that v = lim 0,, 0, e 2(Rn), 0, = 0
(11.6) '">•
[ in the neighbourhood of {xn = 0}, limit in H1/2 (Rn).

56 11. Subspaces of HS{Q). The Spaces HS0(Q)
But applying Lemma 11.1, with X = H1'2^1), Y = JSPfRjT1),
we already know that
f v = lim Vj in #1/2(R"),
(H.7) \Vje9(RXH;H1»Wc71))l
[ y)j = 0 in the neighborhood of {xn = 0}.
The assertion (11.6) follows by truncation and regularization of y>j in
%' (which is permissible in #S(R",-1) Vs). Therefore, the proof of the
theorem will be complete as soon as we have shown Lemma 11.1 to hold.
Proof of Lemma ILL 1) Let v -» N(v) be an antilinear continuous
form on the space if defined by (11.4) and (11.5). Then, if we use
diagonalization and set
w(Jl9x) = (**(t))(A),
we may represent N(v) (according to the Hahn-Banach theorem) in
non-unique fashion by
+ 00 00
(11.8) N(v)=j j(Xf(X,x) + \x\^g(X,x),w(X.x)\wd/t{X)dx,
— oo Ao
where
(11.9) /€l2(Rt;^), geL*(Rt;l)).
If
(11.10) h =,Ft^-1(A/(A,T) + \x\«2 g(X,x)),
we have, in particular
(11.11) heSr'{Rt;X'){Xc Y c X')
(y (Rr; X') = space of tempered distributions taking their values
in X') and
(11.12) N{<p) = (h,<p) V<pey(Rt]X)
{Sf (Rr; X) = space of rapidly decreasing, infinitely differentiate
functions taking their values in X, the parentheses in (11.12) denoting the
antiduality between &"{Rt\ X') and ^(Rr; X); see L. Schwartz [6]).
2) Suppose that N(<p) = 0 for cp e3>{Rt\ X), cp = 0 in the
neighborhood of 0. Then:
h has support in t concentrated at {t = 0}
and therefore (L. Schwartz [1], [6])
(11.13) finite
I £j e X', d{t) = unit mass at the origin.

11.2 A Property of HS(Q), 0 ^ s < \ 57
But then computing !Ft 9l (h) and setting
(11.14) «ft = fcW
and comparing with (11.10), we obtain
(11.15) Xf(X,x) + |r|1/2g(A,r) = £ (ir)',,(*).
finite
But, according to (11.9), we have in particular
1
■^f + \r\^g)eL2(Rt^.l)
1 + \r\1/2
and therefore, for all % e f)lt we have:
(i r)J
y TTT^J'^) (wnere (??./»#) denotes the antidual between
finite 1 + |t|1/2
I !)_! and i^) belongs to L2(Rt).
This is possible only if, for all %, (rjJtx) = 0, therefore r\5 = 0,
therefore h = 0, therefore A/ + |r|1/2g = 0 and then (11.8) shows
that N(v) = 0 y/v. D
Remark ILL The proof of Lemma 11.1 essentially uses the Fourier
transform and the fact that we are dealing with the spaces L2 with
hilbertian range, but points out the role of the exceptional parameter £.
For an analogous result in the case of the spaces Lp, p 4= 2, consult
Lions-Magenes [1] (IV), Theorem 1.1, p. 313. □
11.2 A Property of H'(Q)9 0gs<i
We introduce a function q by
q is infinitely differentiate on D, positive on Q, vanishing
on r of the order of d(x, r) (= distance from x to r), i.e.
such that
(11.16)
lim j, ' =<Z*0 if x0eR
x-+x0 d(x,r)
Such functions do exist, since r is an infinitely differentiable variety.
Theorem 11.2. Assume that Q satisfies (7.10) and (7.11). Let q be
defined by (11.16). Suppose that
(11.17) 0^ s <\.
Then
(11.18) u->q-*u
is a continuous linear mapping of Hs (Q) -» H° (Q) = L2 (Q).

58 11. Subspaces of HS(Q). The Spaces HS0(Q)
If Q = Rn+ = {x | xn > 0}, the theorem holds with q {%) = xn.
Proof. 1) According to the density Theorem 11.1, it is sufficient to
show (the c/s denoting suitable constants)
(11-19) llr'plljiow) ZcAvIbw V?e^(£).
With the help of local maps, to show (11.19) amounts to verifying:
(11.20) K->IIho<r»> ^ c2 \\<p\\HHR^ V? e ®(R"+).
2) In general, Y being a Hilbert space, we define (see Remark 9.5)
(11.21) H°(0, oo; Y) = [//x(0, oo;Y),L2(0, oo;Y)]„, l-6 = s,
where
H^O, oo; Y) ={i)heI2(0,oo;Y), w* e£2(0, oo; Y)},
(oo \ 1/2
j(ii»(oiir+i[»'wii?)^) •
The properties we have seen to hold for the scalar case extend, without
difficulty, to (11.21).
Therefore, in particular (Theorem 9.1), every v e Hs(0, oo; Y) is a
restriction of weHs(R] Y) to ]0, oo[, i.e. satisfying
(1 + | r |s) w e L2 (RT; Y), (w = Fourier transform of w).
Therefore
f Hs(Rn+) c #s(0, oo; Y), Y = ^(R^r1),
(11.22)
[ with continuous injection.
(In fact, we have precisely (compare with (9.6))
Hs(Rn+) =L2(0, oo ; H'iBZT1)) n H'(0, ao;Y)).
3) Consequently, (11.20) follows from the stronger inequality
(11.23) ||r'y||L2(0ia);r) ^ c3 ||y||H.(0i«,;r). V? 6#(]0, oo[; Y).
We shall use the identity
(11.24) p(*) = i/(*) -ie>(*), * > 0, (pe@(]0, +oo[; Y),
where
(11.25) »(*)=lffa(*) -?(£))<*£,
0
oo
(11.26) »(*) = \—v(i)dS.

11.2 A Property of HS{Q), 0 ^ 5 < \ 59
Indeed, if cp e^QO, + oo[; Y), v(x) and w(#) -» 0 as x -» + oo, from
which (11.24) follows, if we verify q>' = v' — w', which is immediate.
4) Then (11.23) will, according to (11.24), follow from the two
inequalities:
(11.27) ll*-SllL2(0.oo;y> ^ *5 llvllH.(0.oo;y).
(11.28) IU"S^|lL2(0,oo;y) ^ ^6 l|yL.(0.oo;y).
Proof of (11.27). According to the definition (11.25) of v, we have:
X
\\v(x)\\2Y^^{\\<p(x)-<p(£)\\2Yd£,
0
therefore
00 00 X
j x-i° \\v(x)\\2Ydx ^ JV*"1 dxj \\v(x) - <p(Z)\\2Yd£
0
oo oo
0 £
oo oo
= fd£J(S + t)-2'-1 \\<p(Z + t)- <p(£) fy dt g
0 0
oo oo
£ J>(2,+1)<*f J II¥>(£ + ')-?>(£) II?<*<.
0 0
from which we obtain the desired result, according to (10.23).
Proof of (11.28). The inequality (11.28) is a consequence of (11.27)
and of
(11.29) ||^S^||L2(0.oo;y)^^ll^-^llL2(0,oo;y) (0 < S < ±) .
This last inequality may be verified as in Lemma 10.1 — and also
can be deduced from it by noting that the transpose of the mapping
X 00
- g(y)dy is g-> — g(f)if;
x t
0 x
therefore (11.29) follows from (10.9), taking <x = -s. D
Theorem 11.2 is completed by
Theorem 11.3. Assume that (7.10), (7.11) and (11.16) hold and that
\ < s ^ 1; then (11.18) is a continuous mapping of Hq(Q) -* L2(Q)
(and the same is true with q(x) = xn, when Q = [x \ xn > 0}).

60 11. Subspaces of H'(Q). The Spaces HS0(Q)
Proof. Since by definition of HS0(Q), 9{Q) is dense in HS0(Q), we
have to prove (11.23) - this time, with \ < s ^ 1.
Instead of formulas (1L24), (11.25), (11.26), we use:
(11 25a) <p(x) =v(x) + wt(x)
X
(11.26a) «M*)= | j»(*)<**
0
oo
(which follows from (11.24), (11.25), (11.26) and | — v(£)d£ = 0 if
We still have (11.27) if s > •£. Finally (11.28) is replaced by
(11.28a) ||*"sWilk2(o.oo;r) ^ c8 IMIhho.oo;*).
which follows from
(11.29a) Is-wllixcojT)-^ c9 ||^~* w|b(0.».r)-
Setting —v = f, this inequality is equivalent to
x
■i-(^hdi)
^ C9 ||^1-S/llL2(0.ao;r)>
||L2(0,oo;Y)
which is the same as (10.9) (setting 1 — s = <x < -£). Q
11.3 The Extension by 0 outside O
Theorem 11.4. Assume that Q satisfies (7.10) and (7.11).
(11.30) u -► u = extension of u by 0 outside Q,
is a continuous mapping of HS{Q) -► Hs(Rn) if and only if 0 ^ s < \.
(11.30) is a continuous mapping of Hq(Q) -► Hs(Rn) for s > \ if and
only if s =)= integer + \.
Proof. 1) According to Theorem 9.4, if s > \, we may define u\r
and u\r. But since u = 0 outside Q, we must have u\r = 0, therefore
u\r = 0, which is not the case (since u -► w|r is a surjective mapping
of Hs(Q)-^Hs-^2(r)).
Therefore, the mapping (11.30) may eventually be continuous only if

11.3 The Extension by 0 outside Q
61
2) We show that the mapping is continuous for 0 ^ s < -£. Since
(Theorem 11.1) the space @(Q) is dense in Hs(Q)f it is sufficient to
show that
(11.31) ||^||h.(r-)^c1||v||ii^) Vq>e®(Q) (0£s<*).
(The case s =^ 0 is obvious; we have equality with cx = 1.)
Via local maps, it is sufficient to show (11.31) with Q = R*+. Using
Sections 10.2 and 10.3, we see that (11.31) will follow from the inequalities:
00
I r<2s+1>dt J \q>(Xl xt + t xn) - <p(x)\2 dx
0 R"
00
g C2 j T<2*+1> dtj \<p(xlt. ..,*,+*,...,*„)- <p(x) |2 dx
0 R^
I for I <> i < n.
(11.32)
The inequalities are obvious for 1 ^ i ^ n — 1. To prove the case
'* = w", it is sufficient to show that
(11.33)
+ 00
jt-<2s+1>dt j \$(x + t) -<?(x)\2dx^
0 — oo
00 00
^c3jt-<2s+1>dtj\<p(x + t) - <p(x)\2dx
0 0
= c3N(<p) V9>e^(]0,oo[).
(Apply (11.33) to xn -► 9?(^', #n) and integrate with respect to x'.) This,
in turn, reduces to showing that
oo 0
f t~i2s+l)dt \\$(x + t)\2dx ^ c4N(<p) (notation of 11.33)
o
or that
i.e. that
(11.34)
jt-^^dtj\<p(y)\2dy^c4N(<p),
00
(y)\2dy^c4N(<p),
which is true, according to Theorem 11.2. D
3) We shall now prove that (11.30) is not continuous when s = |.
Indeed, (11.34) is necessary for continuity; if this inequality would hold

62 11. Subspaces of HS{Q). The Spaces HS0(Q)
with s = \> it would be true for all cp e H1/2(0f oo); but if cp is once
continuously differentiate, with compact support and cp(0) = 1,
(11.34) is not true for s = \. D
4) Finally, the continuity of (11.30) as a mapping of Hs0 (Q) -* Hs (Rn)
for s > \ if s =)= integer + \, follows from the following considerations:
(i) when \ < s < 1, the continuity may be verified as in 2), using
Theorem 11.3 instead of Theorem 11.2;
(ii) for integer s ^ 1, the result follows immediately, since
IIPIIh».(r»> = II^IIh-co) V integer m ^ 1;
(iii) for m < s < m + 1, we use Theorem 9.7; indeed, if cp e @(Q),
we have, according to this theorem, the equivalence of
MU.a» with £ \\D'V
\\p\^m /
As we have seen that cp -► $ is a continuous mapping of Hq (Q) -► Ha (Rn)
for ex * J, 0 ^ (T < 1 (if (T < i, ffg(fl) = #"(£)), we see that
f z ii^iih.-^)) zcJ x ii^vn^-^y
\|p|£m / \\P\Zm J
and since, again according to Theorem 9.7, the first term is equivalent
to ||$||h«(r»)» we finally have
II^IIh-cr-) ^ c2 WvWnw Vpe0(fl),
from which the desired result follows. D
11.4 Characterization of //g(f2)-Spaces
Theorem 11.5. Assume that Q satisfies (7.10) and (7.11).
Let s > \. Then the following two conditions are equivalent:
(11.35) ueHs0(Q)t
ueH5(Q)
dJu
(11.36)
dvJ
= 0, 0 ^ / < s -i.
Proof. We have already seen that (11.35) => (11.36). There remains
to be shown that the converse holds. Via local maps, this amounts to
showing that if ueHs(Rn+), with
dJu
then u e Hs0 (R\
(11.37) ~d7{x'f0) = °' °^'<s-i'

11.4 Characterization of //J(£)-Spaces 63
For the moment, we denote by E the space of u eHs(Rn+) satisfying
(11.37); it is a closed subspace of Hs(Rn+) (according to Theorem 9.4).
We need to show that 2(R\) is dense in E. But, in the same way as
for Theorem 11.1, Lemma 11.1, we note that everything follows from
Lemma 11.2. Let X and Y be two Hilbert spaces satisfying (2.1).
Consider the space of vs such that
f veL2(Rt;X), \r\s v eI2(RT; Y)
(11.38)
| v">(0) =0, 0^/<s-i,
with the norm
(11.39) (l|i'l|y(R,!«+ll|T|*«||ia(R,!r))1/2.
The subspace of cp e 2f (Rf; X), with cp = 0 in a {variable) neighborhood
of t = 0, is dense in the space of v's.
Proof of Lemma 11.2. As for Lemma 11.1, if v-*N(v) is continuous
antilinear on the space defined by (11.38) and (11.39), we have
(compare with (11.8)):
+ 00 00
N(v)= J" /(A/.+ lTl'f.^a^WiT,
— 00 Ao
where
f,geL2(RT;ty.
We introduce h (compare with (11.10)) by
/*=^-l(A/+|T|sg).
Then, if N(q>) =0 for all cp e 2 (Rf; X) vanishing in the neighborhood
of 0, we have (compare with (11.13)):
and since
and
finite
F%h= £ (ixytiAX) (®Zj = rlj)
1 + |t|s
it follows that 0 ^ / < s — \. Then
= E «a)(*)®ft.
0^J<s-l/2
& qih

64 11. Subspaces of HS(Q). The Spaces HS0(Q)
if and only if
jn.wft.JVnr
^-dr < oo,
i.e.
oo
jfoj(X)\\w>Vu+im"-'*/*(*)<«> ie-
(11-40) rje([X,Y]u+ll2)ls)'.
Then, if veL2(R,;X), \x\°v eI2(RT; Y):
0^J<s-l/2
where the parenthesis corresponding to the index "/" denotes the anti-
duality between [X, Y]u+l/2ys and its antidual.
But then, if vJ(0) = 0, 0 ^ / < s — \y we have N(v) = 0, whence
the desired result. D
11.5 Interpolation of /7j(0)-Spaces
Theorem 11.6. Assume that Q satisfies (7.10) and (7.11). Let
st > s2 ^ 0,
sx and s2 #= integer + \. If
(11.41) (1 - 0) sx + 0 s2 * integer + J,
(11.42) [flj (fi), H%2 (^e = < '6)Sl +*a (Q)
(with equivalent norms).
Remark 11.2. Hypothesis (11.41) is essential; as we shall see, the
statement of the theorem is false if, for example, (1 — 0) st + 6 s2 = \. □
Proof. 1) Let us assume for the moment that
| [HZ(Q)iH^(Q)]e = HiQl-e)m(Q)
I if m is an integer, (1 — 6) m + integer + \
Then (11.42) follows. Indeed, choose integer m ^ st, i = 1,2. Then
hv(Q) = ma),HO(Q)]et, (1 _ 0i) m = v
Therefore
[1ft (Q), fl? (#)]* = [[#o (fi). #° (O)].,, [H'S (Q), # ° (fl)]J#
(11.43)

11.5 Interpolation of Hq(Q) -Spaces 65
and according to the reiteration Theorem 6.1, this last space is
[#£(£), tf°(£)](1 -0)01 + 002'
from which we obtain the desired result by applying (11.43) once more.
We still have to show (11.43).
2) First, we show that
(11.44) [HZ (Q), # ° (Q)]9 cz H? ~d)m (Q).
We consider the mapping (11.30), which is a continuous mapping of
flJJ (Q) -» flj (Rn) and of H° (fl) -» #° (R»),
therefore, by interpolation, of
[HZ(Q), H° (Q)]9 -» [#m(R»), #° (R»)]a = ffa-«>»(R»).
Then, if « e [#?(£), fl°(fl)]a> we have:
« = (*)fi> we#(1-8)m(R")
and since w = 0 outside of Q, we have
= 0 on r9 0 ^ ? < s - ±,
therefore
—T = 0 on T, 0 ^ ; < 5 - J,
and therefore, according to Theorem 11.5,
ueH$-°*m(Q)9 whence (11.44).
3) Assume, for the moment, the truth of
Lemma 11.3. // Q satisfies (7.10) and (7.11), then for every integer m
there exists an operator
(11.45) u->Ru,
having the properties:
(11.46) Re&(Hk(Rn);Hk0(G)) V*f 0 ^ k ^ tn, k = integer,
(11.47) R(u) = u MueHk0(Q). Q
Then, by interpolation
Re<?(H(1 -°)m (R»); [ff J (fl), tf ° (fl)]a).
So that, Hue H$ ~e)m(J2) atu?*/ (1 - 6) m * integer + i, the function w
is, according to Theorem 11.4, in Ha~6)m(Rn) and therefore
u = RuE[HZ(Q)tH°(Q)]e>
rom which we have the inverse inclusion of (11.44) and the theorem. D

66 11. Subspaces of HS(Q). The Spaces HS0(Q)
Proof of Lemma 11.3. Via local maps, we are brought back to the
case ((Q = RV'. Set (compare with (2.21)):
m
(11.48) Ru(x) = u(x) -][>kw(*' ~ kx»)>
where the ock's are defined by
m
(11.49) £(-&)-> *k = 1, O^j^m-l. D
The case (1 — 6) st + 0 s2 = integer + \ in Theorem 11.6 is m /ac£
singular, as the following theorem shows.
Theorem 11.7. Assume that Q satisfies (7.10) and (7.11). Let
si > s2 ^ 0> si and s2 + integer -f ^.
(11.50) (1 - 0) st + 0 s2 = fi + i, fi an integer ^ 0.
Iri^o1^), #o2(^)~L is a space independent of st and 6
with (11.50).
Set: [Hs0^Q)fHs^(Q)]d = H^^2(Q). Then
f ffg+1/2 (fl) = (u | « e # g+1/2 (fl), o-^ DpueL2 (Q)
(11.52)
( V^> with \p\ = p (q defined by (11.16))},
the interpolation norm being equivalent to
(11-53) ||h||HSo+1/^)=(ll^llH,-/^)+ I \\Q-ll2Dpu\\2LHQ)) .
\ \p\=i* I
The space H^q1,2(Q) is strictly contained in H%+1/2 (Q), with a strictly
finer topology.
Proof. 1) Choose an integer m ^ max(sx, s2), then 0t from (1 —0t)m
= sif i = 1, 2. According to Theorem 11.6, we have (since st #= inte-
ger + ^: H°0< (Q) = [H™ (Q) ,H°(Q)]ei
and, according to the reiteration Theorem 6.1, we have (with equivalent
norms):
[HS1 (Q), Hi? (Q)], = [flj (Q), H° (%.Wl+Wl.
2) But still according to the reiteration theorem:
[HZ(Q), H°(Q)]e} = [[flj(fi). H°(Q)]XI, [H%(Q),H<>(Q)]X2-]a
if (1 — <x) <x1 4- (X«2 = 03- Choose a^ and <x2 according to
(1 - oct) m = fi + 1, (1 - a2) m = //.

11.5 Interpolation of H5Q(Q)-Spaces 67
Then (1 — a) (xx + a,<x2 = 03 implies oc = \ and therefore (using
Theorem 11.6), we have:
[fl? (fl) , fl? W]. = [#S+ ' W - ffg W] 1/2 •
Therefore we have (11.51).
3) To obtain the characterization (11.52), we shall use the fact that
[H^l(Q),H^(Q)]l/2 = H^1'2(Q).
We come back to the case
J2 = {*|*„>0}, q = xn.
We shall apply Theorem 10.1 and Remark 10.4, as well as the
following fact: define in Hft(Q),
Gt W = group of translations of t on the variable xt, 1 ^ i ^ n — 1,
Gn (t) = semi-group, defined by
f 0 if xn < t
[/(*',*„-*) if x>L
If — At is the infinitesimal generator of Git 1 ^ i ^ nt we have
1=1
and therefore, according to (10.16):
'ueH£+ll2(Q)
(11.54) Metfg^1'2 (£)<*]
jr2\\G„(t)u-u\\2HS(S»dt<co.
The condition on the integral is equivalent to
\t-2 Z [\\Gn(t)Dxu-Dxu\2dx)dt +
o M=*\a /
+ jt-2]jL<(j\Gn(t)Dxu-D>u\idx\dt< oo.
But if ueHg(Q), the second integrals are finite and therefore (11.54)
yields:
ueH&ll2(Q)-
wetfg+1/2(£)
00
j t~2 j \Gn(t) Dpxu -I/Xu\2dxdt< oo
o a

68 11. Subspaces of HS{Q). The Spaces HS0{Q)
We rewrite the integral condition:
00
jt-2dtj\D"u(x',xn + t) -D"u(x)\2dxn +
o a oo t
+ jt~2dt j dx' j\D'u{x)\2dxn< oo.
o r;:" o
The first integral is finite for all u e H$+l/2(Q) and we see that:
ueH£+ll2(Q)
ueHtill2(Q)-
\t~2dt \ dx' j\Dpu(x)\2dx„< oo
yp with \p\=fi.
But this last integral amounts to
I
— \D'u{x)\2dx,
x.
[H*0>0(Q),H%0(Q)]6
whence the theorem. D
Remark 11.3. The same proof, for the case st = integer + \y yields
i^H^-^-^iQ) if (1-0)5l+0s2 * integer* £
( = H<?0-eUl+»*{Q) if (1 -6)sx + 0s2 = integer + \
and the analogous result if only one of the st is in the form
''integer + i". D
Remark 11.4. Each ^(Q) is closed in HSi{Q)\ nevertheless,
[Hs0l(Q),Hs02(Q)]9 is (unfortunately) not always closed in [HSl(Q),
HS2(Q)]d; more precisely, it is closed except when 6 satisfies (11.50).
Remark 11.3 yields the same conclusion starting from two spaces
(HS^(Q)) which are not closed in HSi(Q) ... D
Remark 11.5. Let Q = ]0, T[ and
otf1^) ={u\ueH1(Q)fu{0) =0}.
Then (same proof as above)
(11.55) \JI1(0)^(0)]i,2^oH1o,2(Q)
= {u \ueHl'2(Q)t t-1'2 ueL2(Q)}.
This can be extended to the case of functions taking their values in a
Hilbert space and will be used in Chapter 4, Volume 2. D
Remark 11.6. We shall not go into a general study — somewhat
teratological — of spaces [Hq1 (Q), Hq2 (Q)]e when sx or s2 is equal to
k + \y with k an integer. D

11.5 Interpolation of Hq(Q) -Spaces 69
Remark 11.7. The problem to which we called attention in the
preceding remark is probably also tied to a new characterization of Hq (Q)-
spaces, which we shall only give here for integer s (we shall make use
of this characterization in Chapter 2, Section 7):
Theorem 11.8. Let integer s > 0. Then u e Hq(Q) if and only if
ue@'[Q) and
(11.56) q-s+\«\D«ueL2(Q) V* with |*| ^s
(q defined by (11.16)).
Proof. 1) Let ueHq(Q)) we prove (11.56). Via "local maps" and
"partition of unity", we are led to the following situation: we have
a function v(x',xn) belonging to Hs0(Rn+) and with compact support
in R+. We must show that
(11.57) x;s+j+^DJXnDl,v(x',xn)eL>(Rn+), 0 rg / + \y\ ^ s.
But since veH% (R+), we have that Dyx, v e Hs0~^(Rn+). Thus, we may
apply Lemma 10.1 to the (vector-valued) function:
xn -* w(xn) = D$,v(x',xn) eI2(Rn_1) and to its derivatives;
we obtain
00
J*»-2(,-"'|)ll^«'(«'.*.)llL(R-.)^.^
o
00
<; Cjx;2i'-W \\xnDXnDlv(x',xn)fLHRn-l)dxn
0
00
= C j"*;*-M> + 2 IID,^.^*',*,,)!!^,,.,,^. ^
0
00
£ C2J*-2<—"'•l,+2 \\x„D2xDl,v(x',xn)\\2LHRn.l)dxn
0
oo
= C2 J,;«-lv|)+« WlDZMx'.x.niMR.-'idx.Z
0
oo
^ C-I'lJ*;2 \\xnDx:^Dl,v(x',xn)fLHRn-l)dxn
0
= c-|vl||Z)-IHi));,„||^R„)<+00(
and therefore (11.57) is verified.

70 12. The Spaces H~S(Q), s > 0
2) Conversely, suppose u e 2'(Q) and (11.56) holds. Then, we obtain
u e HS{Q) immediately from (11.56). We show that u may be approached
in HS(Q) by functions of 2(Q) and therefore that u e HS0(Q). Indeed,
let dv(x) be a sequence of functions of 2(Q) such that
dv(x) = 1 if d{x9T) ^~-
v
and
dv{x) =0 if d(x9F} £ —
and
\Dadv(x)\ ^ ——j-j^- (where Ca depends on *);
d(x,ry]
such a sequence exists.
Then dvu e HS(Q) has compact support in Q. We verify that dvu-*u
in #*(£), as v -» + oo, i.e. £>a(<5,w) -» £>"w in L2(J2) for |*| ^ s.
But dvD"u -* D"u in L2(Q); therefore it is sufficient to show that
(11.58) D'<J¥D'«-»0 in L2(£) if |j8|^l, \y\ + |jff| = \*\ ^ s.
Now, DfidvDvu vanishes ii d(xfT) ^ 2/v since |/?| ^ 1, so that,
according to Lebesgue's Theorem, we have (11.58) if we note that, thanks to
(11.56), we have
|D'<JVD>«| ^ -^ |Z)yw| ^ c, e-*+W |Z>'«| e L2(£).
Thus it is sufficient to approach dvu (v fixed) in #s(i2) with functions
of 2{Q)\ and this is possible by the usual method of regularization,
since the support of dvu is in a compact subset of Q. D
Remark 11.8. Denote by Lq-0(Q) (real a ^ 0) the space of functions
ueL2(Q) such that Q~aueL2(Q), with the norm of the graph. Then
[L2e-s(Q)>L2(Q)]e = L2e-Hl-e)(Q)
by definition of [X, Y]0-spaces, since L\-*(Q) is the domain in L2{Q)
of the self-adjoint operator of multiplication by q~s. Since Theorem 11.8
implies that HS0(Q) a L\-S(Q)> we have
(11.59) [Hs0{Q)fL2{Q)]d-c Le2-Hl-e)(Q), integer 5 > 0, 0 < 6 < 1.
12. The Spaces H-S(Q)9 s > 0
12.1 Definition. First Properties
By definition, we set
(12.1) H-(Q) =(#*(£))', 5> 0.
Since 2(Q) iw dense in HS0(Q) (by definition), we have:
(12.2) H-S(Q)^3'{Q). U

12.2 Interpolation between the Spaces H S{Q), s > 0 71
Example 12.1. Definition (12.1) coincides with (7.5) when Q = R"
and if we take definition (7.1). D
Example 12.2. When s is an integer, we have the structure theorem:
Theorem 12.1. Let m be a positive integer. Then every f e H~m(Q) may
be represented, in non-unique fashion, by
(12.3) /= £ D>f„ /f6lJ(fl).
Proof. 1) Through the mapping
u-+{Dpu\ \p\ ^ m}t
the space Hm (Q) is identified to a closed subspace of a product of L2 (£})
and therefore, according to the Hahn-Banach theorem, every
continuous linear form u -► L(u) on Hm (Q) may be represented (non-uniquely)
by
(124) £(«)=£ fgpDpudx, gpeL2{Q).
v ' ' 1*1 *ȣ
2) Now if Z, e(H™ (£}))', it may again be represented by (12.4) and
this time L is uniquely defined by L (q>) V<p e3>(Q)\ but L (cp) = </, cp},
where / is defined by (12.3) with fp = ( — l)|p' gp, the brackets denoting
the duality between ^' (i2) and ^ (i2). Thus, the theorem is proved. D
Remark 12.1. We must be careful to distinguish between the spaces
H~^2{Q) = dual of (Hl'2(Q) = Hl0'2 (Q))
and
(#J/>2 (£))' = dual of Hl0'02(Q)
which, according to (11.52), we may also represent by
(12.5) fe(Hl0/02(Q))'of=f0 + fl, f0eH~*'*(£}), Q"2f1eL2{Q).
Thus, the function
~l-eE(H^02(Q)y V*>0. D
Q
12.2 Interpolation between the Spaces H~s(f))9 s > 0
According to the duality Theorem 6.2 and with Theorems 11.6 and
11.7, we have
Theorem 12.2. Assume thatQ satisfies (7.10) and (7.11). Let s2 > sx ^ 0,
sx and s2 =)= integer + \. Then, if
(12.6) (1 - 6) sx + 6 s2 * integer + \y
we have
(12.7) [#-Sl(£),#-S2(£)]0 = H-(l-d)5l-d52{Q).

72 12. The Spaces H~S{Q), s > 0
//
(12.8) (1 - 0) Si + 6 s2 = \ + ii (integer /a),
then
(12.9) [H-5l(Q),H-S2(Q)] = (tfg0+1/2 (£))'. Q
It seems natural now to "interpolate" between Hs0l (Q) (resp. HSl (Q))
and H~S2(Q). We shall successively examine these points. D
Remark 12.2. The case st = integer + \ will not be discussed (see
also Remark 11.6). D
12.3 Interpolation between Hs0l((i) and ETS2((i)9 5, > 0
Theorem 12.3. Assume that Q satisfies (7.10) and (7.11). Let sx and
s2 ^ 0 and 4= integer -f \. Assume
(12.10) (1 - 0) sx + 0 s2 4= fjL + i <m<* 4= - /* - i (integer // ^ 0).
Furthermore, stiV if st 4= integer + %, we have:
(12.12) [^(fl),^-2^)]^^172^) */ (l-9)si-9s2=^ + i
(12.13) [fl*0« (Q), H-*2 (Q)]e = (Wt1/2 (fl))'
*/ (i - o) sl - e s2 = -/j, - i.
Proof. Theorem 12.3 follows from the preceding results and the
reiteration theorem. First of all, from (2.41), with V=H%(Q), H=H°(Q),
we deduce:
[flj(0),ff—(fl)]1/2=.H»(fl).
Applying the reiteration theorem, we have
(i) if <x < i,
[flj(fl),ff-(fl)]s = U.ffS(Q).H-'(Q)-]0 ,[HZ(Q),H-»>(Q)]ll2]2a
= [H'S(Q),H0(Q)]2x = H%-2*)m{Q) (by Theorem 11.6),
if (1 - lot) m * ft + J, and = tf&1/2(.Q) (Theorem 11.7),
if (1 — 2a) m = ft + \;

12.4 Interpolation between H'X(Q) and H~s*(Q)t st > 0 73
(ii) if a > \t
[HX(Q),H-(Q)]a = [[flJ(fi),ff-(fi)]1/2 [Hm0{Q)tH-™ (i3)]1]2a_1
= [H°(Q), H-m(Q)]2(X.l = H^~2^m(Q) (by Theorem 12.2),
if (1 - 2oc) m * -/* - i, and = (#g0+1/2(£))'
if (1 — 2a) m = — ii — \.
Next, having chosen integer m ^ max(sx, s2), we note that,
according to (i):
WW = [H'S(Q),H-»>(Q)]ai, (1 - 2*x) m = Sl
and according to (ii):
H'S2(Q) = [Hm0{Q))H-^{Q)]0i2t . (1 - 2*2) m = s2.
Then,
[AS' (A). #"S2 W], = [[# o (#), #"" (0)]„, [# o (O). «"" W]J.
= [flS(fl),ff-"(fi)](1_„.1+tal
and the theorem follows by applying (i) and (ii). D
12.4 Interpolation between HSl(£i) and H"S2{Q), 5, > 0
The interpolation between H5l(Q) and H~S2(Q) is somewhat more
delicate than between H'0*(Q) and H'S2(Q).
Theorem 12.4. Assume that Q satisfies (7.10) and (7.11). Let sx and
s2 ^ 0, with s2 + ft + % (integer p ^ 0). We /jav^:
(12.14) [#"(£)> H-S2(Q)]d = ^(1-0)Sl-0S2(i3)
*/ (1 - 0) Si - 0 s2 * -\ - v (integer v ;> 0),
(12.15) [H5*(Q)t H'52(Q)]e = (ffSS1/2(fl))'
t/ (1 - 0)sx -0s2 = -J-y.
Proo/. For the time being, we assume
Lemma 12.1. // Q satisfies (7.10) and (7.11), then for all integer
m^.\\
(12.16) [H»>(Q),H-»>(Q)]l/2 = #<>(£).
Then, choosing m ^ max^, s2), we have, if (1 — 0X) w = sx:
ffs'(G) = [#"(£), tf°(£)]9l = [ff-W.tff-W.ff-tfi)]^],,
= [ff»(0),ff-(fl)],l/2.

74 12. The Spaces H~S(Q), s > 0
If (1 — 02) m = s2 4= (i + i, it follows from Theorem 11.6 and the
duality theorem that
H-°*(Q) = [H°(Q),H--(Q)]1_H = l[H-(Q),H-*(Q)]l/2,H-*(Q)]l^
= [ff"(Q),H-*(Q)]1_9ll2-
Therefore
[H«(Q), H-»(Q)]e = l[H-»(Q), H-">(Q)]ei/2, [H"{Q),H-{Q)]y^l2\
= [#"(£), #-(£>)]■„,
where
-"-s>t + 9('-t),
so that (1 - 2a) w = (1 - 0) sx - 6 s2.
If a ^ \, we have:
[ff»(0), ff-(0)]a = [[ff»(fl), ff-(fl)]0. [A"(fl), ff-(fi)]l/2]2.
= [ff-(fi),fl»(fi)]2a = tf(1-2a)m(£).
If « > i, we have:
[ff"(fl),ff-(fl)]B = {{H"(Q),H-m{Q)ll2, [H'«(Q),H-'«(Q)]1]2x.1
= [H°(Q),H-'»(Q)]2x_l
= ([Ht(Q),H<>(Q)]1_{2a_1)y,
which is equal to (Theorem 11.6)
(H{02«-1)m(Q)y = H{1-2«)m(Q)
if (2<x — 1) m #= \ + r, integer r ^ 0, i.e. if (1 — 0) st — 6 s2 4=
integer — \, and is equal to (Theorem 11.7)
(ffj£2+'(fl))' if (1 - 6) sx - 6 s2 = -* - r. U
Proo/ o/ Lemma 12.1. 1) Consider the mapping rfl = restriction to Q.
This is a continuous linear mapping of, Hm(Rn) (resp. H~m(Rn)) -►
-* Hm(Q) (resp. H~m(Q)), therefore by interpolation, it is a
continuous linear mapping of
[Hm(Rn)tH-m(Rn)]1/2 = H°(R*) - [H"(Q),H-*(Q)]1/2.
Then, HueH0 (Q), its extension w by 0 outside i2 is in H° (Rn), therefore
rQu = uE[Hm{Q)tH-m{Q)]ll2i
therefore
(12.17) #°(£) c [^m(i3),^-(i3)]1/2.
(Note that this inclusion is valid without any regularity hypothesis on
the boundary f of Q.)

12.4 Interpolation between HSl{Q) and H~Si(Q)f st>0 75
Variant: [#£(£), H~m{Q)]l/2 = H°{Q) c [Hm{Q)fH~m(Q)]l/2.
2) The inverse inclusion of (12.17) follows from the following lemma:
Lemma 12.2. // Q satisfies (7.10) and (7.11), there exists, for every
integer mt an extension operator P having the following properties'.
(12.18) Pe&(Hm{Q);Hm{Rn))
(12.19) Pe&(H-m{Q);H-m(Rn))
(12.20) rQPu = u VueH-m{Q).
Then, by interpolation:
Pe&{[Hm{Q),H-m{Q)]l/2', [Hm{Rn),H-m{Rn)]l/2 = #°(R")),
therefore, if u e [Hm(Q), H-m(Q)]l/2, we have: Pue H°{Rn), therefore
yqP u(= u) e H°(Q), from which the inverse inclusion of (12.17)
follows. D
Proof of Lemma 12.2. Via local maps, it is sufficient to construct P
with properties (12.18), (12.19), (12.20) when Q = R\. We define P
by
u(x) if xn > 0,
2m
%<xku (%', ak xn) = u* {x) if xn < 0 [ak < 0),
(12.21) Pu(x)
where the <xks and aks are chosen to satisfy the relations
2m
(12.22) £**«*= 1, -m^j^m-1,
which is possible: first, choose the aks such that
(12.23) det.aJk * 0,
then the ocks as the solutions of (12.22).
Conditions (12.22) for 0^/gw-l imply (12.18), since they
interpret the fact that
dJ dJu±
_^0)=—f (*',0), O^j^m- 1.
oxn oxn
Conditions (12.22) for -m ^ j <^ -1 imply (12.19); indeed, (12.19)
is equivalent to (fP denoting the transpose of P considered as an element
of &{3){Q)\3)(Rn)))
(12.24) fP e &(Hm (R»); ff J (fl)).
But, as can easily be verified, if fe@(Rn), then
(12.25) tPf = f(x)-g(x), xn>0,

76 12. The Spaces H~S(Q), s > 0
where
2m
(12.26) g(*)= £«*"'«*/(*',«*"'*„)
k=l
and then conditions (12.22) for —m^j^ —1 interpret the fact that
dJf dJg
and therefore we obtain (12.24) (applying Theorem 11.5 with s = in).
Property (12.20) being obvious, the lemma is proved. D
12.5 Interpolation between HSl(Q) and (HS2(Q))'
Let sx and s2 be fixed and positive.
The space (HS2 ({?))', dual (or anti-dual) of HS2 (Q), is not necessarily
identified with an "ordinary" function space but is an "abstract"
space. We may identify HSl(Q) with a dense subspace of (HS2(Q))\ in
the following manner. Let u e HSi (Q). Then
v -» \ u
v dx
is a continuous antilinear form on HS2{Q)t say u*. Thus, we have a
linear mapping u -» u* of HSi (Q) -» (#*2 (Q))', which is continuous and
one-to-one: if w* = 0, then
f uvdx = 0 VveHS2(Q)f
therefore Vv e3f{Q)9 therefore u = 0. Identifying w* and w, we have:
HSl(Q) c(HS2(Q)y and ff"(A) is <fcnss. Indeed if we((H'2(Q))')'
= HS2{Q) and w = 0 on #Sl(J2), we have:
jwydx = 0 V<p e #Sl (A), therefore ie> = 0.
Furthermore, note that this identification is independent of sx
and s2. D
Theorem 12.5. Assume that Q satisfies (7.10) and (7.11). Let sx <m^
s2 ^ 0. Wtf have
(12'2?) f^(1-0)Sl-0S2(i3), t/ (1 - 0)sx -0s2 > 0;
[fi..(fl,,,«»(e)n. = j(H.((1.,>„..,!,(fl)),i/(1.9)Si.es2-0
(This time, there is no exceptional parameter such as in the preceding
theorems.)

12.6 Interpolation between Hs0l(Q) and (HS*(Q))' 77
Proof. We use the same type of reasoning as for Theorems 12.3
and 12.4, for example. First, note that, according to (2.41):
[ff-(fl),(ff-(fl))']1/2 = fl»(fi).
Choose integer m ^ max (s1,s2). We have (Theorem 9.6):
Hs> (Q) = [fl*(Q), H° (fl)],,, (l-ei)m = si
= [H»>{Q),[H»>(Q),(H'»(Q)y]l/2]et
= (according to the reiteration theorem)
[Hm(Q),(H-wy\ttll.
By the duality Theorem 6.2,
(H^(Q)y = [Hm(Q),(H-(Q))']1_e2l2
and therefore (by the reiteration theorem)
[fl* (Q), (H* (Q))']e = [H*(Q), (fl-(Q))']a,
*=(l-0)y + *(l-y
if « g i, [A»(fi), (fl*(fl))']„ = [ff»(fl), [fl»(fi), (H">(Q)y]m]2a
whence the first equality in (12.27).
If *>t,
[H»(Q),(H"(Q))']X = [[H*(Q), (H"(Q))']ll2, (H^Q))']^.,
-[H°{Q),(H»(Q))']2a.l
= ([ff"(fl),fl°(fl)]2_2a)' = (#«-»« (fl))',
whence the second equality in (12.27). D
12.6 Interpolation between HfriO) and (H'2(0))'
Theorem 12.6. Assume that Q satisfies (7.10) and (7.11). Suppose st^.O
and sx #= integer + \. Then
// (1 — 0) sx — 0 s2 ^ 0 and #= integer + ^;
(#-((1-0)5l~0S2)(J2))'
#8J1/2(fi) */ (1 -0K-0s2 = i + /*,
integer // ^ 0.
(12.28)
[fl?(fi).(^(fi))1.=

78 12. The Spaces //"*(£), s > 0
Remark 12.3. Theorems 12.5 and 12.6 may be compared as follows:
except for certain exceptional values of the parameter, the interpolated
space between Hs0l (Q) and (HS2 (Q))' is a closed subspace of the
interpolated space between HSi (£}) and (HS2 (£}))' (for the same parameter). D
Proof of Theorem 12.6. By duality we deduce from (12.16):
(12.29) [flj (Q), (#•» (£>))'] 1/2 = H° (Q).
Since st # integer + \, we have:
tfs0' (fl) = [flJJ (fl), H° (Q)]9i, (l-e1)m^s1,
and applying (12.29):
(12.30) H% (Q) = [HZ(Q), [Hm0(Q), (Hm(Q))'] 1/2]9l
= [//J(0),(fl"(fl)y],l/2.
Next
H°> (Q) = [fl» (Q), H° (Q)]92, (1 - e2) m = s2,
therefore, by duality,
(tf^£))'= [tf ° (£),(#-«(£))'] i-»2
= [[fl?(fl), (Hra(^))']1/2, (ffffl(^))']1-92
= [H^(Q),(Hm(Q))']i2,2+l_e2,
which, together with (12.30) (and the reiteration Theorem 6.1), yields
[H% (Q), (H°> (Q)Y\, = [HZ (Q), (Hm (Q))\,
with
If oc ^ \t we have:
[#£(£>), (tfra(£))']„ = [HZ(Q), [HlS(Q),(Hm(Q))']1/2]20i
= [Hm0(Q),H°(Q)]2a
Ha-2*)m(Q) if (1 - 2«) »» * integer + J
#So1/2(£) if (l-2«)w-i + /«
(integer /j ^ 0).
If oc > \, we have:
[HZ(Q), (Hm(Q))']x = ([H"(Q), H°(Q)]2_2x)> = (/?««-»»(0))'. D
Remark 12.4. Lets > 0. rQ (restriction to £) maps tfs(R") -»• #*(£)
and according to Theorem 9.1 it is surjective. Therefore, if
(12.31) Ker, (ra) ={u\ueH° (R"), ra u = 0},

12.7 A Lemma 79
we see that rn is, by passage to the quotient, an isomorphism of
Hs(Rn)IKers(rQ) onto HS(Q) and therefore, by transposing:
(12.32) *(ra) is an isomorphism of (HS{Q))' onto (H5(Rn)IKevs(rQ)y.
But
(^(Rn)/KersM)' =
= {/1/e#-*(R"), </,?>> = 0 Vp6 0(R")f with q> = 0 on CQ}.
Therefore set
(12.33) Has(Rn) = {/ | feH~s{Rn)f f with support in D}.
Then
(12.34) f(rn) is an isomorphism of (HS(Q))' onto Hq5(Rh). Q
Remark 12.5. Since Hq(Q) is a closed vector subspace of HS(Q)
(s > 0), we may identify (through the isomorphism (12.34)) the dual
H~S{Q) of HS0(Q) with the quotient of \rQ) (H'(Q))' = ^S(R") by
the subspace of Hqs(Rh) which is orthogonal to Hq(Q), i.e. by the
distributions /e#ss(Rn) such that
</, rQ <p} = 0 V<p eS(R")f rfly eflj(fl).
Therefore / = 0 in i2 and
feH~s(Rn) = distributions of #~s(Rn) ze**A s«^or* *n T.
Therefore, in short:
ff-(fl) *ffjJ'(R»)/ff;:'(R»).
Also, in this connection, note that the completion of 2{Q) = (space of
2(Q)-functions extended by 0 outside Q) for the topology of H~s(Rn)
may be identified with Hgs(Rn) (and is therefore essentially distinct
from H~S{Q)). Q
Remark 12.6. If sx = integer + \t result (12.29) is still valid if we
replace Hs0l (Q) with HS00(Q), as the same proof shows; indeed, if we use
H& (O) = [Ho (Q), H° (Q)]ei, (1 - 6X) m = sx,
the proof of Theorem 12.6 applies immediately. Therefore, in particular
[^(fll.^W = <-e)Sl"es2(£)
if (1 - 0) sx - 0 s2 ^ 0, * integer + \. D
12.7. A Lemma
We shall give below (Lemma 12.3) a technical result which will be
useful for the study of the regularity of solutions of elliptic boundary

80 12. The Spaces H~S(Q), s > 0
value problems (Chapter 2). In this section, we set:
(12.35) Q = {x | xn > 0}
(12.36) K(Q)
du „ 1
u\u, e#r_1(J2), l^i^n-l,
dxt
where r and n are two integers ^ 1, r — /j, of arbitrary sign. We provide
K{Q) with the norm
/ , V,1 II 5m
3"m
a«?
Hr-M(£)
1/2
||H»--i(0)
which makes it a Hilbert space. We define K(Rn) in the same way.
Lemma 12.3. With K(Q) defined by (12.36), we have
(12.37) K(Q) =Hr(Q).
Proof. The case tfr ^ //" is immediate. Thus, the interesting case
is ((r < fi,\
For the time being, we assume
Lemma 12.4. The space 2(D) is dense in K(Q).
and we show that Lemma 12.3 results from it:
for u e 2(D), we define the function P_u in Rn by
(12.38)
Pu(x) =
\u(x) if xn > 0
£ A,«(*', -jxn) if xn < 0,
U=i
(#' = {«!,.. • ,*„-i})> where the A/s are chosen so that
(12.39)
I(-/)*Aj=l. 0^*^-1.
i=i
(12.40)
Then, we verify that:
f (u -* P u is a continuous linear mapping of 2(D)
[ (provided with the topology induced by K(Q)) into K(Rn)
First of all, thanks to (12.39), we have
(D"u{x) if xn>0
D"Pu(x) =
£ A,/>>(*',-/*»)) if *„<0

12.7 A Lemma
81
for all <x such that |«| ^ p, therefore in particular V# such that \<x\ ^ r
and therefore
(12.41)
Next
d
Hr-l(Rn)
|P_W ||Hr-l(R») + Z
/ "-Ml du II \
\ 1=1 0*1 flr.l(fl)/
(12.42) —Pw = !
I ^U /XT
—-(*) if *„>0,
and we shall obtain (12.40) if we can show that
(v-+Qvis& continuous linear mapping of @(D) (provided
{ with the topology induced by Hr~" (Q)) into Hr~" (R"),
where
iv if xn > 0
I(-/r^(*W*») « *»<o.
For this purpose, it is equivalent to show that Q* (adjoint of Q) is a
continuous linear mapping of H^~r (ft") -> #g~r (fl). But if q> e ^"r (R"),
we have
(12.45) Q* q> = ?(*) -tyjY-lh<p(x', - i*„) (*. > 0)
and
d*(>*9> , a*?)
dxl
^o)=U(x''°{i-i(-ir~i-i)"]
Therefore, thanks to (12.39), —rQ* <p(x', 0) = 0 for 0 ^ k ^ u - 1,
therefore for 0 ^ A ^ // — r, whence the result (12.43) and therefore
(12.40).
Because of Lemma 12.4, we may extend u -» Pju by continuity to
a continuous linear mapping of K (Q) -» K (Rn), which we still denote
by u -» P^.
Now, we verify that
(12.46) K(Rn) = Hr(Rn).

82 12. The Spaces H~S(G), s > 0
We use the Fourier transform; let v be the transform of v eK(R"),
£ = {£', £„} the dual variable of x = {x', x„}; we have:
(12.47) (l + lfl)(l + \Z\Y-1veL2(R")
and
(12.48) |||M- t)eL2(R").
V ' ' (1 + III)""' V '
We have to show that (1 + |£|)r» eL2(RB); since
{i + \s\Y£cl[(i + \?\)(i + \e\rl + \e.n
it is sufficient to show that ||„|r v eL2(R"). But
\i.\rd + mrr ^ c2[\i„\" + \em\'(i + inr-'j s
^c2[if.r + (i + ifi)(i + |{iri],
from which we obtain
llB|r-C2[(TT%^+(1 + iri)(1 + mrl}
from which the result follows according to (12.47) and (12.48).
Thus, if u eK(Q), we have: v = P_u eK(Rn), therefore according
to (12.46) v e Hr (Rn) and the restriction of v to Q (that is u) is in Hr (£3). D
We still have to prove Lemma 12.4.
We shall make use of a new lemma, which is of interest in itself:
Lemma 12.5. Let X be a Hilbert space; for s a positive integer, we
define:
#-*(0,oo;X) =(iro(0,oo;X))'.
Let v satisfy
(12.49) veH~s(0, oo;X), v^eff-'fO, oo;Z), (v™ = ^-jY
(12.50) v^ e #~s(0, oo; X), 0 ^ / ^ ft.
Proo/. 1) The space ^([0, oo[; X) is dense in the space of v's
satisfying (12.49) (with the natural norm). Indeed if L is a continuous anti-
linear form on this space, we may write it
(12.51) L{v) = (w09v) + K,^(k)), w0tw1 eH50{0, oo;X).
We assume that
(12.52) L{v) = 0 Vye#([0, co[;X)
and we want to show that L = 0.

12.7 A Lemma
83
If w-t = extension of wi by 0 for t < 0, we deduce from (12.52) that
®o + (-D*-^! =°-
Then:
(12.53)
But w0eHs(R;X), therefore (12.53) implies
^etf'+^RjX) and wx eHso+k(0, oo; X).
L(v) =(w0 + {-l)ku$\v) = 0.
2) For v e@([0, oo[; X), we define
v(t) if t>0
s+k+l
£ cjv(-jt) if t<0,
(12.54)
ft V (t) =
with
(12.55)
i ;=i
s + fc+l
J=i
(» ^)(*> =
= -7T 7)(W
Then z; -» ^ z; is a continuous mapping of @([0, oo[; X), provided with
the topology induced by H~s(0f oo;X), -» #"S(R,X); next
*;<*>, if * > 0
'l (-j)hCjiP>(-jt)9 if ^0|
l J=i J
and the mapping vik) -» n vik) is continuous, for the topology induced
by #-*(0,oo;X), ->H-S(R;X).
Therefore, through extension by continuity, using 1), we see that
every v satisfying (12.49) is a restriction of nv to ]0, oo[ with
(12.56) n v e H~S(R; X), [n v)™ e ff-*(R; X).
But by Fourier transform in tf, we immediately verify that (12.56)
implies
(7ivY»eH-s(R;X), 1 £/£ k- 1,
and (12.50) foUows. D
Proof of Lemma 12.4. 1) Let u e K{Q). Via regularization in x\ we
immediately see that u is the limit in K{Q) of functions v satisfying,
for example
\veH°(0, oo;#*,), v«~" e#°(0, oo; Hx,)t
(12.57)
v^eH'-^O, oo;Hkx,),
where k is chosen arbitrarily and Hkx> = Hk(Rnx, 1).

84 12. The Spaces H~S(Q), s > 0
2) According to Lemma 12.5, we obtain in particular from (12.57)
that
(12.58) v^ eH'-»(0, oo; #*,), r^j^[x.
3) We define
(»*(*) = *>{*', *» + h), xn > -h (h> 0)
I vh = restriction of wh to Q
and we verify that
(12.60) vh -» v in if (J2) weakly as A -» 0.
In fact, the only thing to verify is that v{f e #r-"(0, oo; #*,) and that
„<#«> _» v<#o in ff-"(0, oo; Hhx.) as A -» 0. But, for cpe2){Q):
W\<p) = (-iriv-^cp(x'txn-h)dx
(where cp{x't xn — h) is extended by 0 for xn < h)t from which the result
easily follows.
4) To show the lemma, it is therefore sufficient to approach, in the
sense of K(Q), with a sequence of functions of @(D), a function v
satisfying (12.57) and (12.58) and which is a restriction to Q of w having
the same properties as v but on Qh = {x \ xn > —h}. Therefore, in
particular
(12.61) w^eH'-'i-h, oo; #*,), r^j^p.
We consider a function 6 = d(xn), C00 on R, and such that 6(xn) = 1
if xn ^ -A/3, 6(xH) = 0 if xn ^ -2A/3. Thanks to (12.61), Ow has
the same properties as v, this time on Rn, and therefore in particular
v = restriction to Q of 0 e K (Rn).
But, through regularization and truncation, 0 = lim^, in K(Rn),
0jE@ (Rn), and by restriction to Q: v = lim^ in K (Q), ^ = restriction
of 0j to Q, e@(D). U J
Remark 12.7. With the same type of procedure, we could show the
following (a variant of the intermediate derivatives theorem of
Section 2.3):
if weL2(0,oo;X), «<"■> e#"*((), oo; Y), then
a"* eH-kJlm(0f oo;[X9Y\J/m). U

12.9 Invariance by Diffeomorphism of HS(Q) -Spaces 85
12.8 Differential Operators on Hs (ii)
Let Q satisfy (7.10), (7.11) and let A be a differential operator of
order N with infinitely differentiable coefficients in D.
We want to investigate how A operates on Hs (Q), with for example
s > 0. This is a simple exercise starting with the preceding results on
interpolation and the fact that
(12.62) Ae&(Hm(Q)\Hm-Ii{Q)) if m is an integer (^0 or <0).
(Moreover, we shall obtain, in the same fashion, the properties of A
on H50(Q)f H'n{Q), H-S{Q), etc.).
Proposition 12.1. We have
(12.63) A e&(Hs(Q)'t H5~N(Q)) if s - N * -v - i, integer v^O
and
(12.64) A e&(Hs(Q)'f(H^-s(Q))') if s - N = -y - \y integer v^O.
Proof. Let ju, be an integer, ju, ^ max(iV, s). We apply (12.62) with
m = n and w = 0. By interpolation, we obtain
(12.65) AeX([H"(Q)9H*[Q)]9; [(H"-»(Q),H-"(Q)]9).
We choose 0 from (1 - 0) = s. The first space in (12.65) is HS(Q)
(definition (9.1)). Through an application of Theorem 12.4, in order to
interpret
[H"-"(Q),H-«{Q)]t,
we obtain the desired results. D
Remark 12.8. We note that (particular case of the preceding
Proposition) :
d
e^(H5(D)]H5-1(D)) for s ^ 0, s * ±,
(12.66)
-^e<?(H^(Q);(Htf(Q))>).
By transposition (for example), we verify that
e^(Hs(D);Hs-1(Q)) Vs < 0.
12.9 Invariance by Diffeomorphism of /75(&)-Spaces
Let Q' be another open set in R" having properties analogous to
those of Qt and let 0 be an infinitely differentiable diffeomorphism
of D onto D'. Then 0 induces, in a natural manner, an isomorphism

86
13. Intersection Interpolation
of HS{Q) onto H'(Q') (resp. HS0(Q) onto HS0(Q[), resp. Haw(Q) onto
^oo (^') if s = integer + £); this can be shown directly, first for integer
s ^ 0, then, by interpolation for real s > 0; finally, by duality we pass
to the case s < 0.
13. Intersection Interpolation
13.1 A General Result
Theorem 13.1. Let 34? be a separable Hilbert space and A0, At two
positive, self-adjoint, commutative operators in Jt. Let D(At) be the
domain of At, provided with the norm of the graph.
We have, V0e]O, 1[:
(13.1) [D(A0)nD(A1),Jf]e = [D(A0)fJf]en[D(A1),^]d
(with equivalent norms).
Proof. Because A0 and Al are commutative, there exists a
simultaneous "diagonalization" of the At's (see Dixmier [1], p. 217).
More precisely, there exists a measurable sum:
e
where d[i{X) is a measure ^ 0 on A0 ^ A00 > 0, Ax ^ A10 > 0, and
a unitary operator ^ of ^ onto f) such that <%(D(At)) = {/|/ef),
A,/el)} = domain of {AJ, the operator of multiplication by Xit and
W(Atu) = Aj(#«) V«eD(^).
Then
*(Z>(4>) n 1)0!)) = {/ | / e f), A0 / s %, X, f e f)},
a condition equivalent to (A0 + Ax) / e f).
Therefore, {A0 + AJ denoting the operator of multiplication by
(10 + Ax), we have
^(D(A0)nD(Al)) = D({^0 + Xl})
and property (13.1) is equivalent to
(13.2) [Z)({A0 + A1}),Jf], = '[Z)({^)}),Jf],n [DtfAj),.*],.
But according to the definition of the spaces [X, Y]d, the first space is
the domain of {A0 + AJ1"0, that is the space of functions /el) such
that (Ao + ^-'/e*.
which is a condition equivalent to
ft-'feh A}-9/ef),
whence the theorem. D

13.3 Example of Application (II) 87
13.2 Example of Application (I)
We shall make frequent use of Theorem 13.1 in the following chapters:
the theorem is useful in situations where we have to work with function
spaces defined on product spaces of the type Rx x RJ' (or Qxx Qy, Qx
(resp. Qy) an open set in Rx (resp. Rjf)), with different properties accord-
ing to the variables.
This is the case for evolution equations, quasi-elliptic equations etc.,
in fact for all applications to non-elliptic differential operators. Here,
we give a very simple example. D
On the space Rx x R$f, we define
(13.3) flj; = {u I [(1 + If |2)"2 + (1 + \rj\2V12] u eL2(R^')},
where
1 f
dtf.i?) = /2g\(.+.')/a exp{-i{x£ + yij))u{x,y)dxdy.
RjcXRy
Note that
r/0,0 _ jjQ _ r2 /pn + n'\
£1xty "~ -"x.y — Lxjl^x,y )*
When st and ax ^ 0, we immediately deduce from Theorem 13.1
that
(13.4) [H'x\fl, Hsx2;ya2]e = ^-e).i+e.2.(i-a)ai+a«- Q
13.3 Example of Application (II)
We now introduce spaces which are to play a fundamental role in
Chapters 4, 5, 6 of Volume 2.
Let J2 satisfy (7.10) and (7.11), and let T be its boundary. In the
space Rx x Rf, we consider the cylinder
(? = i3x]0,T[,
with lateral boundary:
I = JTx]0J[.
The spaces H**(Z), <x,p^0.
As for the spaces Hs (£2) we may first define the spaces H*'P (T x R,)
and then H"'P(Z) by restriction to Z.
By analogy with (13.3), we define
(13.5) ^(fxR) = {u | u eI2(Rf; tfa(^))> |T|' weI2(Rt; #°(r))},
where
+ 00
w(r) = -T=r e~[txu(t)dt.
\J2n J

88 13. Intersection Interpolation
It is a Hilbert space for the norm
(NllL(R,;iw)) + II \r\Pu\\2L2(Rr.Hoir)))1/2.
(We shall see in Chapter 4 of Volume 2, why these spaces are
indispensable for the study of evolution equations, and the properties of traces
of these spaces.) D
We define H*'P (Z) as the image of H*'P (71 x Rf) under the mapping
r£ = "restriction to Zy\ provided with the corresponding quotient
norm. Note that H°'°(Z) = H°(Z) = L2(Z). U
Here is an equivalent definition.
Consider the space Hl,0(Z) = L2(0,T; H1 (r)) as a subspace of
Y = L2(Z) for the measure dZ = dTdt; considering H1,0(Z) as the
space X in Section 2, we see that
(13.6) H1,0 (Z) = domain of A0, positive self-adjoint operator in L2 (Z).
In fact, Ar being the Laplace-Beltrami operator on r, we have:
(13.7) A0 = (-ArY'2
and more precisely:
(13.7a) A0u(t) = (-Ar)1/2 (u(t)) for almost all t e]0, T[.
Next, we consider the space
u\ueV{S),-?-eL2{Z)\;
ot \
then, as above:
(13.8) H0tl(Z) = domain oi Alt positive self-adjoint operator in L2 (Z)
and in fact it can be verified that
(13.9)
/ d2 y/2
Ax = I - + 11 , with, as boundary conditions for
d2 du(0) du(T)
- 4- 1, the conditions = = 0.
dt2 dt dt
Then, we have:
(13.10) H*'P(Z) = D(AZ)nD(A{), 0^«,j8^1. D
For ^(fxR), we shall take
(13.11) /li = l 2 + M ' defined ^y Fourier transform,
and then we shall have the result analogous to (13.10) for allot, (i^O. Q

13.3 Example of Application (II) 89
As for Theorem 13.1, we can verify that, Va^/Jj ^ 0:
(13.12) [H*l'^(rxRt)tH*2-fi2(rxRt)]e
_ jjd -0)*i +0«2. (1 -0)01 +602 (P xR)
We extend this result to the case wehre Z replaces fxR (by using the
extension operators of H^(I) -» fl^(fxRr)). D
Now, we introduce (because we shall need spaces of this type in
Chapter 4):
' 0H"-P(£) ={u\uEHa'p(I)t u(.,0) = 0}
fixed /? > \.
(13.13)
This definition has meaning since according to Theorem 4.2, if /? > \,
t -* u (•, t) is a continuous function of
[o, T] -> iH«(r),H°(r)]1/2e = flw<i-l/^)(r)>
so that (13.13) has meaning and defines a dosaJ subspace of H"'P(Z).
We have
Proposition 13.1. Let <x,(i > 0, /? > \. Then
(13.14) [o^"'^), H°(I)]6 = ffU-«>«'U-W(jP), f/ (1 - 0) fi < \.
Proof. Let
H$fi(£) = closure of D(I) in #«•'(£)
{. dJu dJu 1
«l«efl»-^,—(.,o) = —(.,r) = o, o^/</?--
We have
and therefore
(13.15) [#?'(£).ff°(23J9 c [o^'"(^).^°(^)]fl <= [fl-'^.ff0^],.
In (13.15), the last space equals #«-•>«. u-«W(jp). The first (proof
analogous to those of Theorems 11.6 and 11.7) is equal to
#a-ewi-a>/r(2:) if (1 _ 0) ^ + integer + i
(and to the strict subspace H^6)a-ll2{Z) of /#-eu>1/2(D
= H^-n"1'2(S), if (1 -0)0 = }),
whence (13.14), since
fli1""-"-"'!^ = H(1-e)xA1-e)^(n if (1 - 0)0 < f D

90 13. Intersection Interpolation
13.4 Interpolation of Quotient Spaces
Let X and Y be two Hilbert spaces as in Section 2 and
(13.16) N = closed vector subspace of X and of Y.
Let n be the canonical mapping of X -► X* = XjN (and of Y -► Y9
= Y/N); n is a continuous linear mapping of X -► X#, Y -> Y#,
therefore, by interpolation
(13.17) ne<?{[X,Y]e;[X',Y-]e).
Since n is obviously a continuous linear surjection of
[X, Y], - ([X, Y],)/tf,
we have
Proposition 13.2. L^ X and Y be two Hilbert spaces satisfying (2.1)
and let N be defined by (13.16). Then
(13.18) ([X, Y]e)/N c [X/N, Y/N]d, 0 < 6 < 1,
(m'tfA continuous injection). D
The following theorem gives a very restrictive sufficient condition
for equality in (13.18) (but which will be useful in Chapter 2).
Theorem 13.2. Let X and Y be two Hilbert spaces satisfying (2.1)
and let N be defined by (13.16). Then, if N is finite-dimensional, we have:
(13.19) ([X, Y]e)IN = [X/N, YlN]e, 0 < 6 < 1.
Proof. Let z1} . . ., zv form a basis for N, chosen to be orthonormal
in Y. For all u e Y, we define
V
(13.20) Ru = u -Y, (u>zi)Y*i'
i = i
We have: R e&(X; X) n J^(Y; Y) and # = 0 on AT, therefore,
R* denoting the quotient mapping:
R'e&(X';X) n&(Y"f Y),
therefore, by interpolation
R-eJ?([X-,Y']e;[X,Y]e).
Thus if u9 e [X9, Y9]e, we have
R9 u9 e [X, Y]e and n R9 u9 = u9 e ([X, Y]e)IN,
whence the inverse inclusion in (13.18) and the theorem. D
Remark 13.1. The preceding arguments are evidently general, as
long as N is finite-dimensional; neither the hilbertian structure, nor
the particular construction of the interpolation spaces came into play. D

14.1 General Result 91
Remark 13.2. For Chapter 2, we shall also need the following "dual"
viewpoint: let
IiV* = closed vector subspace of Y' (X and Y are not identified
with their duals; therefore Y' c X').
We define
(13 22) \{X>N*}={u\ueXf(u,z*> = 0 V^eJV,}
|{Y;iV*} = H *< e Y, <**,**> = 0 V** e AT*}.
They are closed vector subspaces of X and Y.
Theorem 13.3. Let X and Y be two Hilbert spaces satisfying (2.1) and
let N# be defined by (13.21) and finite-dimensional. Then (with notations
analogous to (13.22))
(13.23) {[X,Y]e;N*} = [{X;N*}f{Y;N*}]e, O<0<1.
Proof. Let z'lt. . . ., z'v form a basis for N+t orthonormal in Y', and
let z!,..., z„, be the elements of Y defined by
the brackets denoting the duality between Y and Y'. If N denotes the
space generated by zlf . . ., zvf we have the decompositions:
X = iV + {X; NJ, Y = N + {Y'.N*},
which brings (13.23) back to Theorem 13.2. D
14. Holomorphic Interpolation
14.1 General Result
Let us again take up the setting of Section 2, with the two Hilbert
spaces X and Y.
The space jf(X, Y).
We denote by J^(Xt Y) the space of functions
z->f{z), 2 = f + i77, O^f^ l,rjeR,f(z)e Y
having the following properties:
[ / is a continuous and bounded function of the strip
(14.1) {O^f ^ l}-» Y
I and a holomorphic function of the open strip {0 < f < 1} -> Y,

92 14. Holomorphic Interpolation
(the holomorphic property is equivalent to the scalar holomorphic
property:
z -» (f(z)f y)Y is holomorphic in 0 < f < 1, Vy e Y),
If (in) e X and r\ -> / (i r\) is a bounded continuous function
of R„ -> X (therefore /(i q) e ^(R„; X)),
(14.3) /(l + i*?)e^(R„;Y).
In general, we set
llgll«(R;Z) = SUp||g(q)||z
and provide Jf?(X, Y) with the norm
(14.4) ||/||JrUir) = max(||/(ii?)|| 11/(1 +i^)IU<R,;n)-
Thanks to the classical three-line theorem, we see that J^(Xt Y),
provided with the norm (14.4), « a Banach space.
Theorem 14.1. Let X and Y be two Hilbert spaces satisfying (2.1).
For 0 < 6 < 1, the mapping
(14.5) /->/(0)
o/ ^f (X, Y) -> Y « w /arf a continuous linear surjection of tf (X, Y) ->
- [x, y],.
TAe norm on [X, Y]e is equivalent to the norm
(14.6) ||«|||[x.ne = infll/IUx.y). /(*)=a.
Remark 14.1. We may therefore also use (14.5) to define [Xf Y]e as
the image of Jf?(X, Y) under this mapping: this definition is valid for
Banach spaces and even for locally convex spaces; this point will be made
more precise and applied in Section 14.2. D
Remark 14.2. We may also use traces to define interpolation spaces
(for example, by applying the notions introduced in Section 10). We
would obtain different spaces in general, but which all coincide in the
hilbertian case (and only in this case).
Proof1. We use spectral decomposition (see Riesz-Nagy [1], J. Dix-
mier [1]), which allows us to define Azf z e C, if A is positive self-adjoint
in Y. We take A to be an operator such that D(A) = X (see Section 2).
1) Let feJt?(XtY). Consider
(14.7) g(z) =A~zf(z), 0£f £ 1.
1 This proof, which predates the general introduction of [X, Y]0-spaces, was
communicated to us by N. Aronszajn in 1958.

14.1 General Result 93
This function is holomorphic in 0 < f < 1; we have:
(14.8) g(i??) = /l-1V(i'?)6^(R,;^),
(14.9) ||g(i7?)|| ^ ll/M) ll&(R„;*)-
Then
g(l +ir,)=A-l-»f(l +iri)e&(Rn;X),
since /l"1"^ e^(Y; X) and since \\A"1-iriyiYiX) ^ constant = clt
we have:
(14.10) g(l+ifi)ea(R„;X)
and
(14.11) |g(l+i?)|| II«(R„;X)'
According to (14.8), (14.10), the function g takes its values in X
(since g is a continuous bounded function of 0 ^ f ^ 1 -► Y, a
holomorphic function of t) < f < 1 -► Y and continuous and bounded,
taking its vates m X, for f = 0 aw<£ f = 1). Therefore
g(0)eX,
therefore
/(9)=/l0g(e)eZ)(/l1-0)
and
&(R„;*)> ll?(l +i»»)IU(R,;r));
from which, according to (14.9), (14.11), we obtain:
(14.12) ||/(0)||
2) We still have to show the surjectivity of (14.5). Let a e [X, Y]Q
= D{Al~e). Define
(14.13) /(*) =Az~ea.
This function is in Jf (X, Y) and
ll/lljrw.r> ^ c4 II«IIdui-«)-
Since f(d) = a, we obtain the surjectivity — and, in addition, we
have constructed a continuous linear right inverse a -► / of [X, Y]e -►
-►^f(Z, Y) — which completes the proof of the theorem. D
Remark 143. In the preceding definition of [X, Y]0-spaces, the
hypothesis that the functions r\ -► f(irj) (resp. /(f + ir])) be bounded
in X (resp. Y) is wo£ essential. For example (and what is to follow is
also not the maximum degree of generality) consider the space
*„,&• Y)

94 14. Holomorphic Interpolation
of functions z -+ f(z) having the same properties as / e JP (X, Y), but
With l/MllxgCe'Hi,
ll/(f + iy)\\r ^ CeKl"l, fixed arbitrary K > 0, 0 ^ f ^ 1.
We define a natural norm on Jf?np(X, Y) by
ll/ll*<x.» = max[supe-Kl"l ||/(i^) ||x, supe"*'"' ||/(1 + iJ?) ||r]
and we define
[X, YJa.exp = space described by f(6) as / describes Jf?exp(X, Y), with
the norm of the quotient by the kernel of f -* f(0).
Then:
Proposition 14.1. We have
[*,Y]0,exp = [X,Y]0,
(where, as usual, the equality is understood with norm equivalence).
Proof. Clearly Jf{X, Y) c JF9xp(X, Y) and therefore [X, Y]0 c
c [X, Y]0fCxp. Conversely, let a e [-X", Y]0(Cxp; therefore, there exists
feJeexp(Xt Y) such that f(0) = a. But we introduce
g(z) =exp(z-0)2f(z).
We easily verify that g e 34? (X, Y) and since g(6) = f(0) = a, we
see that a e [X, Y]e and therefore [X, Y]0>exp <= [X, Y]0.
Corollary 14.1. We s^7/ o&tam £/^ sam^ spaces [X, Y]d by starting
with functions z -+ f(z) of polynomial growth in X for z = itj and in Y
for z = C 4- i r\ • D
14.2 Interpolation of Spaces of Continuous Functions
with Hilbert Range
We first state Remark 14.1 in precise terms.
Let X, Y be two Banach spaces such that
t X a 0, Y a 0, with continuous injection, where 0 is a locally
I convex topological vector space.
(We do not assume that X c Y.)
Let X + Y be the space of elements x + y, x e X, y e Y, provided
with the norm
ll«llx+f = inf (11*11* + \\y\\f),
x + y = a
which makes it a Banach space.

14.2 Interpolation of Spaces of Continuous Functions 95
We define 3? (X, Y) = space of functions z -* f(z) such that
(compare with (14.1), (14.2), (14.3)) z -* f(z) is a continuous and bounded
function oiO£l;£l-*X+Y and a holomorphic function of 0 < £ < 1 -»
-» X + Y, with
/(i,)e*(R,;#), /(l+i,)e*(R,;?)
(the definition of 3$ (R^; X), X a Hilbert space, extends immediately
to the case ^(R^; I),Ia Banach space); we provide it with the norm
max(sup||/(iq)||jf> sup ||/(1 +iq)||y) = H/lljrtf.fi,
which makes it a Banach space.
We again define [X, Y]d as the space described by f(0) as / describes
J^(Xt Y) and provide it with the (quotient) norm
HI[*,Y]e = inf 11/lljrtf.fi.
/(0) = a
which makes it a Banach space.
We verify, exactly as for the Hilbert case, that these spaces have
the interpolation property; let Xlt Yx be & second couple of Banach
spaces (with properties analogous to X, Y) and rc eJS?(Z; JfJ n
c\£(Y\Yx)t then
TiE^dX^eilX^Y^). D
We consider again the Hilbert couple X, Y, and introduce
f X = C°([0, T]; X) = space of continuous functions of
(14.14) < [0,T] -» X, with the norm sup \\f(t) \\x (which makes it a
( Banach space) teLo'Ti
and in the same way
(14.15) ? = C°([0,T]; Y).
Theorem 14.2. Let X, Y safts/y (2.1) and let X, Y be defined by (14.14)
and (14.15). Then:
(14.16) [X,Y]e = CO([0,T];[X,Y]g),
with equivalent norms.
Proof. The proof follows the same line of arguments as the proof
of Theorem 14.1. Let f eJt?(X, Y). Then, as may be easily verified,
A-*f(z)=g(z)ejr(X;Y)
(indeed, setting (A~z <p) (t) = A~z((p(t)), we see that
A~*g&{X- Y), /T1-(*eJ&?(Y;.X")).

96 14. Holomorphic Interpolation
Then g(0) = A~ef(0) eX = C°([0, t];X) and therefore
/(0) = Aeg{6) eC°([0, T];/)^1-0)) = C°([0, T]; [Z, Y]0).
Conversely, if a e C°([0, T]; [X, Y]e), then Az~ea = f(z) is in
^f(Z, Y) and /(0) = a, from which the theorem follows. D
Remark 14.4. In general, for a given Banach space E, let Lp(0, T\ E)
be the space of (classes of) strongly measurable functions / taking their
values in E, and such that
~~ " *\\LPW,T;E)'
IT \1,p
(14.17) M ||/(0\\pEdt\ =11/11,
Lp(0, T] E) is a Banach space for the norm (14.17) (see Bourbaki [1]).
Then, with the same proof as above, we obtain
[L'(0, T;X),Lp(0, T; Y)]d = LP(0, T; [X, Y]e).
14.3 A Result Pertaining to Interpolation of Subspaces
Let X and Y be two Banach spaces such that
l X <=. 0, Y <=. 0, with continuous injection, where 0 is a
(locally convex topological vector space.
We do not assume that X c Y.
We shall now define subspaces of X and Y in a way which occurs
frequently in the applications, as we shall see in Chapter 2, in particular.
For this purpose, let W be a locally convex topological vector space;
let
(14.19) de&(&; W),
(14.20) #*(resp. «0 be a Banach space, with <F C !F, <^ C !F,
[in particular, we may take 3C = <& = {0}). Define
(14.21) (X)dtX = {u\ueX,due3r},
(14.22) (Y)d,s, = {w|weY,3we^}.
Each of these spaces is provided with the norm of the graph (for
example
IMI(*)3.* = IMI* + lld«W,
which makes it a Banach space.
Our problem is to compare the spaces
P,.f.(Y),,f], and p:,Y],)8
the spaces [ , ]e being understood in the sense of the definition in
Section 14.2.

14.3 A Result Pertaining to Interpolation of Subspaces 97
We shall make the following hypotheses'.
[there exist Banach spaces X, °3J such that:
i) X c X c V, ®J c°M aVf
ii) 3eif(I;f)nif(y;^),
(14.23) {iii) there exist 9 e & (X; X) n & (<&; Y) and
rey(f;f) n &($;&)
such that
Theorem 14.3. ,4ssww* *Aa* (14.18), (14.19), (14.20) ani (14.23) ar*
satisfied. Then
(14.24) [(X)a>ar, (Y)8>,]9 = ([X, Y]„)a,[ar>^, 0< 0 < 1
(with equivalent norms).
Proof. 1) In (14.24) we always (i.e. without (14.23)) have the inclusion
of the first space in the second. Indeed, clearly
[(Xh.sc, (Y)d.#]. «= [X, Y]9-
On the other hand (we did what was necessary for that):
deX((X)M;ar)nJ?((Y)0.9;&),
therefore, by interpolation
de^([(X)0,x,(Y)0t9\t;[ar,<sqt),
from which our assertion follows.
2) Conversely, let ae([X, Y]e)d,igrt91e.
This means that there exist /e Jf(X, Y), geJ^{Xy <&) such that
(14.25) * = /(»), da = g(0).
We define h by
(14.26) h(z) =f(z) -9df(z) + 9g(z).
We shall verify that
(14.27) heJr((X)M(Y)M),
in a moment.
Then
h(0) =f{6) -9df(0) + 9g(0) = a -9 da + 9da = a,
therefore a e[(X)dtx, (Y)dty]e, from which the inverse of the
inclusion in 1) and the result follow.

98 15. Another Intrinsic Definition of the Spaces [X, Y]q
There remains to show (14.27). But h(z) eX + Y, is holomorphic
in z with values in X + Y. Furthermore
dh(z) = df{z) - d$df{z) + d$g{z) = (using 14.23), iii)
= *(*)+'(*(*)-3/M);
so that dh(irj) = g(irj) + r(g(irj) — df(irj))\ but g(i^) e #", therefore
r g (i 77) also belongs to 9C and df(irj) e & (since 3 e J£? (X; «3f)), therefore
r(df{irj))e3F and
3A(ii2)e«(ar).
In the same way dh(l + iq) e^(^), whence (14.27). D
Examples of applications of the preceding result are given in
Chapter 2, Section 7.
15. Another Intrinsic Definition of the Spaces [X, Y]e
In general, if X and Y are two Banach spaces satisfying (14.18),
we set, for every a e X + Y and for all t > 0:
(15.1) K(t,a;X,Y)= inf (||a0||J + t2 \\ax \\2)1/2} a0eX, axeY.
It is then possible to define new interpolation spaces by considering
the set of as for which the function t -► K(t, a; X, Y) has various
properties (belongs to Lp(0, oo; t* dt), etc.).
In this way, we have
Theorem 15.1. Let X and Y be two Hilbert spaces satisfying (2.1).
Then
(15.2) [X, Y]e = {a | a e Y, rie+ll2) K(t, a;X, Y) e L2(0, oo)}.
Furthermore, the norms ||«|cx.h« «w<^
,00 ,1/2
l||a||J+Jr<29+1)XM;X,Y)art)
are equivalent.
Proof. We start by showing the formula
(15.3) K(t9a\X, Y)2 =t2(A2{A2 + t2)-1a,a)Y
if X = Z)(/t) (see Section 2), with \\u\\x = \\Au\\Y.
Indeed
#(*,a;X,Y)2 = inf (\\A a0\\2Y + *2 \\a - a0\\2Y).
a0eD(A)

16. Compactness Properties 99
The Euler equation for this minimization problem is
(15.4) (A2 4- t2) a0 = t2 a, a0 being the element reahzing the optimum.
Then a0 being given by (15.4), we have:
K(tta\Xt Y)2 = \\Aa0\\2Y + t2 \\a0\\2Y + t2 \\a\\2Y - 2t2Re(ata0)Y
and since according to (15.4), ||/la0||y 4- t2 \\a0\\l = t2(afa0)Y) we
have:
K(tfa]Xt Y)2 = t2(afa- a0)Y = {atA2a0)Y
= (a,A2t2{A2 + t2)~l a)Y, whence (15.3).
Therefore
00 00
jri2e + 1)K{ttatXfY)2dt = j(t^-2d)A2{A2 + t2)-1 a, a)Y dt.
0 0
But (see Riesz-Nagy [1], Dunford-Schwartz [1], or use the spectral
decomposition as in Section 2, assuming Y to be separable):
00 / 00 \
(15.5) !ta-mA2(A2 + t2)-1dt = cA2(1-nL= fj^-^-ds],
0 ^ 0 '
so that
00
(15.6) \ ri2d + 1)K{tfafXf Y)2dt = c\\Al'e a\\2,
o
In fact, we have to be somewhat more precise: we have (15.5) for
example in D(A2)f therefore (15.6) for aeD(A2) for example. Then
we pass to the completions.
The theorem follows from 15.6. D
16. Compactness Properties
The following result will be used in a fundamental way in Chapter 2.
Theorem 16.1. Assume that Q satisfies (7.10) and (7.11). Let seR.
Then, for every e > 0, the injection
HS{Q)-*HS-£{Q)
is "compact.
Proof. 1) It is sufficient to show the theorem for s > 0. Since then,
by passage to closed subspaces, the injection of HS0(Q) -► Hs0~e(Q) is

100 16. Compactness Properties
also compact and therefore, by transposition H~s+e(Q) -+ H~S(Q) is
compact, from which we obtain the desired result.
2) Therefore, fix s > 0. There exists (see proof of Theorem 9.1) an
operator p:
u -+ puf
a continuous linear mapping of Hs (Q) -► Hs (Rn) such that
(16.1) p u = u a.e. on Q,
(16.2) p u has support in a fixed compact set Kf Q cz K.
(For (16.2) it is sufficient to truncate after having extended by
reflection.)
Let un eHs(Q) and un -► 0 in HS(Q) weakly. We must show that
un -► 0 in Hs~e(Q) strongly. However,
(16.3) vn = p un -* 0 in Hs(Rn) weakly
and since the operation "restriction to Q" is a continuous mapping
of Hs~e(Rn) -» Hs~e(Q)f it is sufficient to show that (16.3) implies:
(16.4) vn-+0 in Hs-£(Rn) strongly,
or, vn denoting the Fourier transform of vnt that
(16.4a) Xn = j(\ + \Z\)2is-£)\vn(£)\2d£iZrJ for n^n(rj).
But
Xn= j (l + |f|)-2£(l + lfl)2sl^(f)|2^ +
+ J (1 +|f|2(s-£)|^(f)l2^,
therefore
(16.5) Xn^(l +M)-2£J(1 + \e\)u\*K(£)\2de +
+ (l+M)2(s-£) J |0„(|)|2<f|.
According to (16.3), t>„ remains in a bounded set of HS(R"), therefore
(16.5) implies:
(16.6) Xn g c,(l + M)-2e + (1 + M)2(s"£) J |0„(£) |2^.
if ISM

16. Compactness Properties 101
Choose M so that cx (1 + M)~2e ^ r\\2 and (16.4a) follows if we can
show that
(16.7) J |fl"(g)|2ig^2(l+M)2('-rt f°r n = n{^-
But, thanks to (16.2), vn has support in K\ therefore if 6 denotes a
function of @(Rn), we have:
(16.8) vH(S) =(vn>0exp(-27zix£)),
where the brackets denote the duality between Hs(Rn) and H~s(Rn).
But when |f | ^ M, 0 exp( — Inixt;) belongs to a relatively compact
set of H~s(Rn) and therefore (vn, 0 exp( — 2n'\ x £)} -> 0 uniformly for
So that, by (16.8), z5„(f) -> 0 uniformly for ||| ^ M, therefore
J vn(£)\2d£->0 and (16.7) holds. D
Remark 16.1. The preceding proof does not economize the hypotheses
on Q. On the one hand, the result still holds if the boundary r is Lip-
schitzian (see Adams-Aronszajn-Smith [1]).
On the other hand, the result may still be true if Q is not bounded
but "sufficiently small" at infinity. (Note that the result is false for
Q = Rn).
We may introduce, in general for s > 0, the open sets Q having
property (Cs): they are the open sets such that the property of
Theorem 16.1 holds. This notion indeed depends on s. D
Remark 16.2. We again consider the general setting of [X, Y]0-spaces.
Theorem 16.2. // the injection X -> Y is compact, then the injection
(16.9) [X9Y]$l-+[XtY]$2, dl<d2 (0<8<<1)
is compact.
Proof. We consider (see Section 1) an operator A, positive and self-
adjoint in Y, such that
X = D(A).
Since X -> Y is compact, the operator A~x is compact in Y and
the spectral decomposition of A is given by the eigenfunctions w,\
iA Wj = kjWj, j = 1,2,...
A,>0, A,- +«>;
choosing the functions Wj so that
{wjtwk)Y =dhJt

102 16. Compactness Properties
the Wj form a complete orthonormal system in Y. Then
(16.11) X = D(A) =[u\u ^XjWj.Zti \*j\2 <+<*]
and
(16.12) [X, Y]9 = [u\u =txJwJ>Y,^l~e) \*j\2 < + oo).
The property to be demonstrated follows without difficulty. D
Theorem 16.2 shows that in order to verify Theorem 16.1 it is
sufficient to consider the case "integer s'\ However, this does not simplify
the situation appreciably. D
Remark 16.3. We also have the following result, which is very useful
for obtaining a priori estimates (see Chapter 2):
Theorem 16.3. Assume that Q satisfies (7.10) and (7.11). Let slf s2
and s3 be given in R and
(16.13) sx>s2>s3.
Then, for all rj > 0, there exists a constant c(rj) such that
(16.14) \\u\\HsHQ) ^ rj \\u\\B.lia> + c(rj) \\u\\B.M}9 VueH5>(Q).
Because of Theorem 16.1, this last theorem is a consequence of the
following general result:
Theorem 16.4. Let X, Y, Z be three Banach spaces such that
llcYcZ
(16.15)
(the injection X -* Y is compact.
Then, Vrj > 0, there exists a constant c{rj) such that
(16.16) Wuh^nWvWx + cWWuWi VueX.
Proof. Suppose (16.16) does not hold. Then for given rj > 0, there
exist
une X and cn -* 4- oo
with
IKIly ^rj\\un\\x + cH\\un\\z.
Introducing vn = uj\\un\\x, we have
(16-17) \\vn\\Y^ri + cn\\vn\\z.

17. Comments
103
But since \\vn\\Y ^ (constant) \\vn\\x = constant, there results from
(16.17) that
(16.18) IKIIz^O.
Now, since ||v„||jf = 1 and X -► Y is compact, we may extract a
subsequence vJ} which is strongly convergent in Y and (according to
(16.18)) necessarily tends to 0; thus || Vj \\Y -► 0, which contradicts (16.17)
and the theorem is proved. D
17. Comments
The spaces HS(Q) (integer s ^ 0) were introduced by Sobolev [1].
The spaces HS(Q), non-integer s > 0, were introduced by numerous
authors and via numerous methods. We have seen that all reasonable
definitions coincide (at least when Q has a sufficiently regular boundary).
The spaces HS(Q), for integer s < 0, were introduced by Schwartz [4]
in connection with the works of Garding [1] and Vishik [1] on the
Dirichlet problem.
Analogous spaces may be constructed by replacing L2(Q) with
LP(Q), p 4= 2 (and even Orlicz spaces). The corresponding spaces, also
introduced by Sobolev, loc. cit., are generally denoted Ws (ii) or WS>P(Q).
Theorems of the type discussed here are valid in the case of the spaces
WS>P(Q), p 4= 2, but with serious additional difficulties; for example
it is known (J. P. Kahane, E. M. Stein) that for the case p 4= 2, the
method of holomorphic interpolation gives a different result than the
method of interpolation by "traces". For p #= 2, we therefore have
different families of spaces (Sobolev spaces WS>P(Q), Besov spaces
Bs>p{Q)t Lebesgue spaces or Bessel potentials HS>P(Q), for arbitrary
real s; for the definitions, see, for example, Magenes [3]) which coincide
for p = 2; we refer the reader to Aronszajn-Smith [2], Aronszajn-
Mulla-Szeptycki [1], Aronszajn [4], Baiocchi [1], Besov [3], Berezanski
[3], Besov [1, 2], Calderon [1, 2], Gagliardo [1, 2], Krein-Petunin [1],
Lions-Magenes [1] (III), (IV), (V), Lions-Peetre [1], Magenes [3], Nikol-
skii [1,4], Nikolski-Lions-Lizorkin [1], Peetre [12], Stein [1], Shamir [2],
Slobodetski [1], Taibleson [1, 2], Uspenski [1] and to the bibliographies
of these works. We shall limit ourselves to the "L2" situation in this
volume and in Volume 3 to situations which may be deduced from it
by "passage to the inductive or projective limit'' and where it is
unnecessary to consider the Lp-theory for p #= 2.
The HS(Q)-spaces are our ''basic tools" for the study of "elliptic"
boundary value problems which we take up in Chapter 2; for boundary
value problems, it is clearly fundamental to define the values of the
functions on the boundary, and this is why it is so important to study

104
17. Comments
traces in HS{Q). This is our essential aim in Sections 1 —5 and 8, where
the theory of traces in HS(Q) appears as a particular case of a more
general theory (which, by the way, is no more difficult to present than
the particular case); we follow the account of Lions [9].
This leads in a natural way to the theory of interpolation of linear
operators (Sections 5 and 6), which plays a fundamental role in the
sequel. The properties in Section 6 are simple particular cases (we limit
ourselves to the Hilbert case) of the results of Lions-Peetre [1]
pertaining to a generalization of trace spaces in the form of "averaged spaces"
(see also Lions [14, 19]). They are also simple particular cases of the
results of Calderon [2, 3] pertaining to holomorphic interpolation,
introduced by Calderon, loc. cit.f S. Krein [3] and Lions [15], which
we briefly present in Section 14. (Our aim in this chapter has not been
in any way to present a complete theory of interpolation. Aside from
the articles just cited, the reader can consult the works of Gagliardo
and of Peetre mentioned in the bibliography, as well as Aronszajn [3],
Aronszajn-Gagliardo [1], Krein-Petunin [1], Krein-Petunin-Semenov [1],
Semenov [1, 2]).
The trace theorems in HS(Q) are due to Aronszajn [2], Prodi [1, 2],
Slobodetski [1] and other authors.
The extension method given in Section 3.2 is due to Hestenes [2]
and Lichtenstein [1] (see also Babitch [1]). Another method (using singular
integrals) is given in Calderon [1] under weaker hypotheses on Q (but
the extension depends in an essential way on the order s of the Sobolev
space). An extension valid for all positive s, when Q has a C°°-boundary,
was constructed by Seeley [3]; the differentiability hypotheses on Q
have been considerably weakened by Adams-Aronszajn-Smith [1].
The density result (Theorem 8.1) is still valid under much more
general hypotheses on Q (see Gagliardo [1]).
Section 10 follows Lions [11], in which further results can be found;
applied to Lp-spaces, the results of this section reproduce a result of
Gagliardo [2].
Lemma 10.1 is known as the inequality of Hardy-Littlewood-Polya [1]
(Section 3.30).
The interpolation results of Sections 11 and 12 follow Lions-Mage-
nes [1] (II), (III), (IV); in particular, in (IV), the somewhat. ..
dangerous question of the distinction between [Hq(Q), H°(Q)]l/2 and
1/2 was resolved, the first space not being a closed
vector subspace of the second. The presentation given here is simpler
than the one followed in Lions-Magenes, loc. cit.; on the whole, we
adopt the presentation of Grisvard [4] (who systematically uses
formula (11.25), already used, for different purposes, but still in Sobolev
spaces, by V. P. Il'in [1]). Another simple presentation of this result

17. Comments
105
is due to S. Jones (personal communication). For Theorem 11.8, often
used by numerous authors, see for example Kadlec-Kufner [1].
Concerning Lemma 12.2, see also Baiocchi [4]; for an extension
valid for HS(Q), s ^ 0 or 5 < 0 (which extends the result of Seeley [3])
see Geymonat [4].
The result of Section 14.3 is due to Baiocchi [5]; a generalization
is given in Baouendi-Goulaouic [1].
The function K(tt a) was introduced and used systematically by
Peetre [8]; see also a presentation of the theory in Goulaouic [1],
Chapter 1. Another point of view, different, but connected with K(tta),
is developed in Golovkin [1].
The determination of "air' interpolation spaces between two Hilbert
spaces is given in Foias-Lions [1], to be completed with Peetre [10] and
Goulaouic [1].
Also useful for the applications to partial differential equations, are
the Sobolev spaces which bring into play derivatives of different order
according to the direction (non-isotropic Sobolev spaces); particular cases
of this situation will be met in Chapters 4 and 5, Volume 2 (where the
time and space variables play different roles). We have not taken up
the systematic study of this question, for which we refer the reader to
Baiocchi [1], Besov [4], Besov-Kadlec-Kufner [1], Besov-Il'in-Lizorkin
[1], Cattabriga[2 —4], Cavallucci [2, 3], Garding-Malgrange [1], Grisvard
[4], Hormander [6], Itano [1], Jones [1], Kree [1], . . ., [4], Lions-Mage-
nes [1] (I), Lizorkin [1], Malgrange [1], Nikolski [2, 3], Pagni [2, 4],
Pini [10, 12], Solonnikov [2], Ramazanov [1], Uspenski [2], Volevich-
Panejach [1], etc.
Similarly, it is useful to introduce weighted Sobolev spaces in the
applications (we introduced the weights t" in (10.5), but they appeared
only as a "tool"); we shall meet such spaces (the spaces 5(Q)f among
others) in Chapter 2 and we shall establish interpolation properties for
these spaces (Chapter 2, Section 7).
A systematic study of weighted Sobolev spaces is not attempted;
the reader may consult Besov-Kadlec-Kufner [1], Grisvard [1], Gey-
monat-Grisvard [2], Lizorkin-Nikolski [1], Morel [1], Necas [2],
Nikolski [1], etc.
We also call attention to the results of Baouendi [1], where the
weights are used in an essential way.
For a study of Sobolev spaces, isotropic or not, and their variants,
systematically using the theory of approximation by entire functions
of exponential type, see Nikolski [5]. One may also (see Yoshikawa [1])
use the fractional powers of operators (see Komatsu [1, 2]).
Using the Fourier transform, we may also define the spaces Hs(x)
of order varying with x\ these spaces come up in a natural manner in

106
18. Problems
the theory of pseudo-differential operators; see Vishik-Eskin [2], Unter-
berger-Bokobza [1].
In Volume 3, we shall meet situations for which it would be
interesting to interpolate "between" spaces without norms; we do not study
the corresponding theory here, for it might well fill ... a volume. The
reader can consult, aside from the last chapter of Lions-Peetre [1],
the works of Deutsch [1] and Goulaouic [1]; see also Girardeau [1].
18. Problems1
18.1 As we pointed out in Remark 11.6, we have not studied the
spaces [Hq1 (Q), Hq2 (Q)]e when at least one of the s/s is of the form
integer + £. The same is true for the case mentioned in Remark 12.2.
18.2 A study as complete as the one presented in Sections 11
and 12, for the spaces WS'P(Q), BS'P(Q), HS'P(Q) (p * 1, 2, oo) remains
to be done, especially concerning the exceptional parameters.
We call attention to the following question: Be,p(Rn) being defined
as a trace space (or an averaged space) between W1,p(Rn) and W0,p(Rn)
= Lp (Rn), and setting
B'e'p{Rn) =(£°'(R»))',
is the space Lp(Rn) an interpolation space between B6l*p(Rn) and
B~62'p(Rn) for 0 < 0t < 1, or not? Does interpolation between triplets
(instead of pairs) of spaces help to solve problems of this type?
18.3 If WS/(Q) = {v\veWs'p{Q)>yjv = 0, 0 ^ / < s - 1#>},
does this space coincide with the closure of <2)(Q) in Ws'p(D)t for all
values of s? In this text, we prove that the answer is yes for p = 2,
and in Lions-Magenes [1] (V) for p =)= 1, oo and s =)= integer + \/p.
18.4 In Chapter 2, Section 6, we shall see how 3s (Q)-spaces, of
the same type as Hs (Q), but with weights on the boundary, are introduced.
The interpolation between these spaces is studied in Chapter 2,
Section 7. Analogous constructions and results pertaining to SStP(Q)t
which are defined like the spaces Ss (Q) but replacing L2 by Lpf
p #= 1, 2, oo, would be of interest.
1 Other problems pertaining to the theory of interpolation will be met in the
following chapters. We call attention only to problems directly tied to the
applications of interpolation to partial differential equations, and not to "general
questions" connected with interpolation (for example, of the type: when does
there exist a Hilbert space or a hilbertizable space of interpolation between two
Banach spaces?).

18. Problems
107
Likewise for the compactness properties of the injection of SStP{Q)
into Ss-e>p(Q), e>0.
Also, interpolation between spaces of the type "weighted Ws,p"
is not completely clear yet.
18.5 Interpolation of Subspaces. Remark 11.4 emphasized one of
the main difficulties of the use of interpolation: the interpolated space
between closed subspaces is not necessarily a closed subspace in the
interpolated space.
It would be of great interest to obtain criteria allowing to affirm
a priori that, except for certain values of the parameters, the
interpolated space is closed.
From the point of view of applications, here is one situation in which
we meet this problem: let H^(Q) be the subspace of Hm{Q) made up
of the elements u e Hm (Q) such that
BjU = 0 on r, O^j^v,
where the B/s are differential operators of order ms < m. Do we have:
[HZ(Q),H°(Q)]9
= {v | v e H(l-9)m{Q), Bjv = 0 on T, if ms < (1 - 6) m - £}
for (1 — 0) m 4= integer + \1
The answer, is yes if the B/s are normal operators (Grisvard [8],
who also investigated the case of the exceptional parameters and studied
the analogous problem for WmtP(Q)); see Chapter 4, Volume 2,
Section 14.5.
Does the result still hold if the B/s are not necessarily normal?
(see also Fujiwara [2]).
Of course, problems of this type exist for Sobolov spaces which bring
in a number of derivatives different according to the directions (which is
useful for evolution equations — see Chapters 4 and 5 in Volume 2 —
and for quasi-elliptic equations).
Likewise, it would be very interesting to dispose of other criteria
than the one in Section 14.3 (due to Baiocchi [5], and which is very
useful).
18.6 Similarly, it would be useful to extend the criteria given in
Section 13 for the interpolation of intersections,
18.7 In a Banach space E, let A be an operator which is the
infinitesimal generator of a semi-group G(t).

108
18. Problems
Consider the space of functions ui (i = 1, 2) such that
f uieLp(0}oofD(A))}
du,
t«—LeLp(0,oz:E)
at
and
du2
Aux + = 0.
dt
What space does [ux (0), u2(0)} describe?
This problem is solved in Lions [16] for the Hilbert case.
The preceding problem is tied to trace problems of the following
type: let u e (Hm(Q))N be such that Du = 0, where D is a differential
system: what space is described by {yj uk}, 0 ^ / ^ m — 1,1 ^ k ^ N?

Chapter 2
Elliptic Operators, Hilbert Theory
For Sections 1—5, with the exception of 5.4, only the definition
of HS(Q)-spaces, for s of arbitrary sign, and the trace theorem of
Section 8.2 are required from Chapter 1. Section 8 may be skipped on first
reading.
1. Elliptic Operators and Regular Boundary Value Problems
1.1 Elliptic Operators
On the space Rn, let
(1.1) A(D)u = £ apD*u
be a linear differential operator of order I with constant coefficients;
we associate to it the polynomial in f = (f x, . . ., fn) e Rn (characteristic
form of A):
(1-2) 40(fl= I *PZP>
\p\ = i
where f* = fp &... f* for p = (plt.. .tpn).
Definition 1.1. The operator A is said to be elliptic if
(1.3) 40(f)*0 VfeR", f*0.
We have
Proposition 1.1. For n > 2, every elliptic operator is of even order.
Proof. Let f and f be two linearly independent vectors in Rn;
consider the polynomial A0(g 4- rf) in the complex variable r.
We have
A0(S + r^') = t< A0(?) + t'"1 At(e,?) + • • • + A,(f)
where At(l;tg') are polynomials in f and f.

110 1. Elliptic Operators and Regular Boundary Value Problems
The equation in r
(1.4) Ao(S + r?)=0
does not have any real roots, when for fixed f', f runs through the set /,
the complement of the straight line in R" joining {0} to f'. Since A0 (f') =)= 0
and is independent of f, the roots of (1.4) depend continuously on f e /.
Furthermore since, for n > 2, / is connected, we see that the number
of roots of (1.4) (of course, each root being counted with its multiplicity)
with imaginary part > 0 (resp. < 0) is constant as f runs through /;
we denote this number by m (resp. I — m).
For f e/, we observe that — f e/ and that
Then:
(number of roots of ^40 ( —f 4- rf') with imaginary part <0) = / — m =
(number of roots of A0(l- — rf) with imaginary part <0) =
(number of roots of <40(f + t£') with imaginary part >0) =m,
therefore I — 2m, D
Proposition 1.1 no longer holds for n = 2: the Cauchy-Riemann
/} /}
operator 4- i is elliptic in R2 and of order 1. However, if
dxt dx2
the coefficients of A are real we see that Proposition 1.1 is also true for
n = 2.
1.2 Properly and Strongly Elliptic Operators
Although for the general properties of solutions of the equation
A u = /, Definition 1.1 of ellipticity is sufficient (see for example
Hormander [6] and Section 3 of this chapter), from the point of view
of the theory of boundary value problems, the property on the number
of roots of (1.4) with positive imaginary part, mentioned in the proof
of Proposition 1.1, is very important (see Section 8). For this reason,
we shall assume, from now on, that the order of A is even (I = 2m)
and introduce
Definition 1.2. The operator A defined by (1.1), with I = 2mf is said
to be properly elliptic if it is elliptic and if for every linearly independent
couple of vectors f and f belonging to R", the polynomial A0(l- -f rf)
in the complex variable x has m roots with positive imaginary part.
The proof of Proposition 1.1 yields
Proposition 1.2. Ifn>2, every elliptic operator is properly elliptic. Q

1.3 Regularity Hypotheses on the Open Set Q 111
A remarkable class of properly elliptic operators is formed by the
strongly elliptic operators:
Definition 1.3. The operator A defined by (1.1), with I = 2m, is said
to be strongly elliptic if there exists a complex number y and a constant
<x > 0 such that
Re(yA0(£))>*\£\2m VfeR". D
Remark 1.1. There exist properly elliptic operators which are not
strongly elliptic; for example, in R3:
a4 a4 a4 /a2 d2 \ d2
dx1 dx2 ox3 \dxl ox2] dx3
Now, let Q be an open set in Rn and assume that the coefficients ap
in (1.1) are functions defined on Q (or D). Then the characteristic form
(1.2) will also depend on x; we shall denote it by A0(x, f) and we shall
denote the operator A by A (x, D). We shall say that the operator A
is elliptic, properly elliptic, strongly elliptic in Q (or D) if for each
x e Q (or D) the operator A (x, D), considered as an operator with
constant coefficients ap (x), is elliptic, properly elliptic, strongly elliptic.
We shall also say that A is uniformly strongly elliptic in D if there exists
a complex number y and ot > 0, independent of xf such that
Re(yA0(x,£))^ot\£\2m Vf e R" et V*e£. D
Remark 1.2. If the boundary jT of Q is connected, the ellipticity of A
in D and the continuity of the coefficients ap in D are sufficient to
guarantee that A is properly elliptic in D, if the condition on the roots
of A0 (x, f + t f') is satisfied for only one point of r and for one couple
of linearly independent vectors f and f.
1.3 Regularity Hypotheses on the Open Set Q and the Coefficients
of the Operator A
Let Q be an open set in R". Since we want to study boundary value
problems for the operator A in the spaces HS(Q), for arbitrary real s,
we shall impose very strong regularity conditions on Q and on the
coefficients of A . We shall be able to use these conditions for all s, but
we could, for each fixed s, impose more general conditions which depend
on s (see Section 8).
We assume Q to be a bounded open set in R", with boundary r, an
(n — Yj-dimensional infinitely differentiable variety\ Q being locally on
one side of T, i.e. we consider D to be a compact variety with boundary
of class C00, the boundary being r.

112 1. Elliptic Operators and Regular Boundary Value Problems
We assume the operator A to be of order 2m and to have infinitely
differentiable coefficients in U\ we write A in the form
(1.5) A u = A (*, D) u = £ (1 -)W D"(apq(x) Dqu)}
with
(1.6) aP9e®{D).
Then, the characteristic form of A is
A0(x,£)= Z (-l)" «,«(*) £P+€. where £p+q = f*1 + Pl . . . fr+q».
\p\.\q\=m
We assume A to be properly elliptic on i3. Under the given
hypotheses, we obtain that A is uniformly elliptic on D, i.e. there exists a
constant c > 0, independent of xf such that
(1.7) c-1|f|2w:g \A0(x,S)\ ^c\£\2m \fxeD and V£ e R".
1.4 The Boundary Operators
The aim of this chapter is to study boundary value problems for
the elliptic equations:
A u = / in Q,
BjU = gj on r,
where the B/s are certain differential "boundary" operators in suitable
finite number and / and gj are given (we formally denote problem (1.8)
by {A.B}).
However, it is well known for the simplest classical cases (for example
A = — A) that we can not arbitrarily assign the operators Bj and obtain
a "well-posed" problem. Therefore, we must introduce certain
admissibility conditions with respect to the operator A on the operators Bj. This
is what we shall do in this section by introducing some definitions
whose role will become clear in the sequel (see, in particular, Section 8).
Let Bj, 0 ^ / ^ v — 1, be v "boundary" operators defined by
(1.9) Bj(p= £ bjh(x)Dhcp,
\h\*mj
with
(1.10) bjhe®{r)
mj being the order of Bj. More precisely, Bj cp denotes the operator
<p-» Z bjh(x)yo{£>h(p),
\h\Zmj
cp being a function defined on D for which yQ{Dhcp)t the trace of Dhcp
on r, may be defined either in the classical sense or in the sense of
the trace theorem of Section 9.2 of Chapter 1.
(1.8)

1.4 The Boundary Operators 113
We consider a subset J\ of r and introduce
Definition 1.4. The system of operators {#/};=o is a normal system
on rx if
a) Z bJh(x) f* * 0 V* e J\ and Vf 4= 0 and normal to T at x,
\h\=mj
b) m, 4= mt for / 4= i.
Now, assume that y = m\ then, we have
Definition 1.5. The system {Bj}™^1 covers the operator A on J\ if for
all x e rit alltj e Rn, not equal to zero and tangent to T at x, and all f' eRn,
not equal to zero and normal to T at x, the polynomials in the complex
variable r: ]T bJh(x) (f 4- xl-')*1, j = 0,. . ., m — 1, are linearly in-
\h\-mj m
dependent modulo the polynomial Y\ (x — xt (x, f, £'))> where xt (x, f, f)
i = l
are £/^ roofc of the polynomial A0(x,g + tf) z^/& positive imaginary
part.
We also impose the conditions
(1.11) %^2w— 1, / = 0,...,m— 1
on the operators £,.
We summarize the hypotheses which we shall use to obtain the
final results of the theory:
1) The operator A is properly elliptic in D and has infinitely dif-
ferentiable coefficients in D)
2) there are m operators By,
3) the coefficients of Bj are infinitely differentiate on J1;
4) the system {jB^}7=o « normal on J1;
5) the system {jB^}7=o covers the operator A on J1;
6) the order ms of B5 is ^ 2m — 1.
If these hypotheses are satisfied we shall sometimes refer to
problem (1.8) as a regular elliptic problem.
We insist on the fact that, for intermediate results, only some of
the hypotheses 1), . . ., 6) will be used. Thus, for example, in Green's
formula (see Section 2) hypothesis 5) is unnecessary; see also
Section 8.4. D
Remark 1.3. Among the systems of operators {Bj} which satisfy
hypotheses 1), . . ., 6) for every properly elliptic operator A, there is
the system of Dirichlet conditions
Bj = yjt j = 0, 1, . . ., m — 1,

114 2. Green's Formula and Adjoint Boundary Value Problems
dJ
where ys = , with v normal to r, oriented towards the interior
of Q. dvJ
Problem {A, B} is then called the Dirichlet problem for the operator A.
Remark 1.4. It is interesting to note that there exist strongly elliptic
operators and normal systems of "boundary" operators which do not
cover them (see Schechter [2], Appendix I).
Remark 1.5. The definitions given in this section are all invariant
under infinitely differentiable homeomorphisms of the open set Q.
2. Green's Formula and Adjoint Boundary Value Problems
2.1 The Adjoint of A in the Sense of Distributions
or Formal Adjoint
With the operator A still given by (1.5) with (1.6), we denote by A*
the operator defined by
(2.1) A*u= £ (-\)^Dp(aqp(x)D9u).
|p|.|fl|£m
A* is often called the formal adjoint of A: actually it is the adjoint
of A in the sense of distributions on Q, for we have
(2.2) f Auvdx - f uA*vdx = 0 Vutve@(Q).
It is easy to verify that A is (properly) elliptic if and only if A* is
(properly) elliptic.
2.2 The Theorem on Green's Formula
Let {Ft}tZo be a system of v differential boundary operators defined
by
Ft 9= I fiH(x)Dh<p,
with/fft(*)e^(r).
Definition 2.1. The system {-FjJ=i is a Dirichlet system of order v
on rt (subset of r) if it is normal on rx and if the orders mi run through
exactly the set 0, 1, . . ., v — 1, when i goes from 0 to v — 1.
Theorem 2.1. Let A be the operator defined by (1.5), with (1.6), and
assume it to be elliptic, let {jB^}7=o be a normal system on r given by
(1.9), with (1.10), (1.11). It is always possible to choose, non-uniquely,
another system of boundary operators {5j}7=o normal on r, the S/s
having infinitely differentiable coefficients on r and being of order fij ^
^ 2m — 1, such that the system {B0, . . ., Bm_lt 50, . . ., Sm_1} is a

2.3 Proof of the Theorem 115
Dirichlet system of order 2m on r. Having made this choice, there exist
2m "boundary" operators Cjf Tjf j = 0, . . ., m — 1, uniquely defined,
having the properties'.
a) the coefficients of Cj and Tj are in @{T)\
b) the order of Cj is 2m — 1 — /Ltj and the order of Tj is 2m — 1 — m5;
c) the system {C0, . . ., Cm_lf T0, . . ., Tm_l} is a Dirichlet system
of order 2m on r,
such that the following Greens formula holds:
(2.3) \Auvdx-\ u A* v dx
Q Q
m-l m-1
= Z \ SjuCjV da — Yj BjuTjV da
j=o j! j=o j!
for all u and v e @ (Q).
2.3 Proof of the Theorem
1) Via "local maps" and "partition of unity" we are brought back
to the case of a half-ball. More precisely, there exists a finite covering
of r by open sets 0t, i = 1, . . ., Nf such that, for all i, there exists an
infinitely differentiable diffeomorphism 0t of Oi onto the unit ball in Rn,
o = {x\\x\< 1},
such that the image of Ot n Q under 0f is a+ = {x \ x e a} xn > 0}
and the image of <9t n J1 is dxa+ = {x \ \x\ < 1, xn = 0}.
Via the diffeomorphism Gt (arbitrary fixed i) the operator A is
transformed into an operator <stft with infinitely differentiable
coefficients in (T+u31(T+, and we immediately see that j/f is still
elliptic; the system {Bj}J=o is transformed into a system of operators
{&j,i}7=o with infinitely differentiable coefficients in dxa+ and which
is normal on dxa+.
If we recall that for ut v e @(D) we have formula (2.2), then we see
that, by using {0J and an appropriate partition of unity, we may
consider the problem under the following hypotheses:
in the half-ball a+f we define a differential operator in the form
(from now on, we shall denote the point x eRn by x = (y, t), with
V = (yi.....y»-i)eR""1 and *eR):
(2.4)
^"= Z ap{ytt)Dl'D^u1 p = (P',pn), f = (pl} . . ., pu_x).
\p\Z2m

116 2. Green's Formula and Adjoint Boundary Value Problems
We assume that the coefficients ap of s/ belong to @>{a+ u 3x0*+)
and that si is elliptic in a+ u diO+.
We also define a system of operators {&j}7=o in the form
(2.5)aj<p= X bJh(y)Dhy'Dht»(p, h = (h',hn)t A'= (*i. • • •• *.-i)
1*1 *«j
normal on c^o^ and with 6j7l e@(d1a+) and m, ^ 2m — 1.
From the above we immediately deduce that we can find operators
{Sf}™Io with infinitely differentiable coefficients in dla+t of order
jLLj ^ 2 m — 1, forming a normal system on dxa+ such that
{^0O, . . ., 2Sm_x, «/0, . . ., Srm_x}
is a Dirichlet system of order 2m on dlo+.
It is sufficient to take
such that the numbers ms and //,, / = 0, . . ., m — 1, run through the
set 0, 1, .. .,2m - 1.
2) At this stage, we prove two lemmas before proceeding with the
proof of the theorem.
Lemma 2.1. If {-F/}S=o and {-F/}$=o #^ ft^o Dirichlet systems of
order v on dla+ and with coefficients belonging to ^(dla+)t then
(2.6) Fj = ^A'JSF's,
j j = 0,...,v-l,
(2-7) F'j = Y,AJsFs>
s = 0
where Ajj and Ajj are non-vanishing functions on 3 iff, belonging to
@(diG+), and Ajs and Arjs (s =)= /) are tangential differential operators
(i.e. consisting of derivatives with respect to yx, . . ., yn_A only) of order
j — s, with coefficients belonging to @(diO+).
Proof. It is sufficient to verify (2.7) for Fj = DJt. Indeed, if {Fj} is
a Dirichlet system of order v, we may write F) in the form
(2.8) F'j = £e'JtDl
i=0
where the &j/s are non-vanishing functions, infinitely differentiable
on diG+ , and the Qjt's, i =)= /, are tangential operators of order ^ / — /,
with coefficients belonging to 2(di<s+). Then, if (2.7) holds for Fj = D{,
that is if
(2.9) Dl-ZQuF., j = 0,...,v-l,
s = 0

2.3 Proof of the Theorem 117
where the 0js's have analogous properties to those of Ajs, we have
Fj = ie,JlDtt = Yde'Jli0is FS = ^AJSFS, j = 0,...,v- 1,
i=0 i=0 s = 0 s=0
where An = O'jj 0jj is a non-vanishing, infinitely differentiable function
J
in dla+ and Ajs = £0^ <Pte is a tangential operator of order ^/ —s,
with coefficients in 2(dxct+).
Therefore, we only need to prove formula (2.9). It holds for / = 0.
By induction, we assume that it holds for / < k and we verify that it
holds for / = k. But {-F/}J=o being a Dirichlet system, we may write Fk
in the form k
Fk = Z6kiDl„
i = 0
where 0ki has the same properties as &'ki. Therefore
O^D) = Fk - £ 0MD\ = F» - 2) ©„ I 0isFs
i = 0 i = 0 s = 0
k-1 /fc-1 \
s=0\i=s J
from which we immediately deduce (2.9) for / = k. □
Lemma 2.2. // {-F/}J=o is a Dirichlet system of order v and with
coefficients belonging to ^(51cr+), then, for every system {(pj}vjZlo of
functions belonging to @(d1cf+)f there exists a function v(y,t)e
e@(o+ u 31o,+ ) such that
FjV = <pJ} / = 0, 1,. . ., v - 1.
Proof. Thanks to Lemma 2.1, we have
(2.10)
s = 0
Di = i0JSFs,
s = 0
with the properties we have already pointed out for A'Js and 0js; we
also have
j
(2.11) YJAfJS0si = dji> O^i^j^v- 1.
s = i
Since the functions ys = £ $si 9^ are infinitely differentiable on
i = 0
d1a+t there exists a function v e^(a+ ^j d1cf+) such that
£f^ = ^s> S = 0, . . . , V — 1.

118 2. Green's Formula and Adjoint Boundary Value Problems
Then, according to (2.11), v also verifies
FjV = YJA'jscPs = i,Aj.£ 0si<pt = £ (t^.*..) = <pj. □
s = 0 s = 0 i = 0 i = 0 \s=i J
3) We return to the proof of the theorem under the hypotheses of 1).
Let u, v e @(o+ u di<f+) vanish in a neighborhood of
d2(r+ = {(y^) I yf + • • • + y--i = i,t = o};
then, integrating by parts first in y and next in t, we obtain
(2.12)
\{^u)vdydt= £ (-1)1"'1 f^«(y,0D;'(«p(y.<)«'(y.'))^y
= X (-1)^ ju(y,t)D"(ap(yJ)v(y,t))dydt +
<y +
2m-1
2m-l
= juj**vdydt+ £ j [DJ,u(y,t)N2m_jv(y,t)]t = 0dy,
\p\^2m
2m-l
+
where
d\<3 +
s/* v = £ (-l)lpl Dp(apv) (formal adjoint of s/)
\p\^2m
and
(2.13) N2m.Jv= £ (-l)1'*-'-1 Dp-'-1 D',\a,(y,f)v(y,t)).
\p\^2m
From the ellipticity hypothesis onj/, we easily deduce that the system
{■^2m-j}j=0
is a Dirichlet system with infinitely differentiable coefficients on d1a+.
Now consider (see 1)) the system {BS0, . . ., @m-i,<SP0, • • ->^m-i}>
of order 2m — 1 on d1a+; in order to simplify the notation, we denote
it by {i^sJsSo1- Then, according to Lemma 2.1, we have
(2.14) Di = fj0Js^s> j = 0>...,2ni- 1,
s = 0
where the 0js's have the properties of Lemma2.1. We denote by 0%
the formal adjoint of 0js (adjoint in the sense of distributions on di<f+),
i.e. the "tangential" differential operator defined by the identity
(21.15) j 0js(pxpdy= j cp0%y)dy, Vcp, y e 2(d1a+).

2.3 Proof of the Theorem 119
From (2.12), (2.13), (2.14), we deduce
(2.16) f(s/u)vdydt- [ u^*vdydt
a+ a +
2w-l 2/n-l
= f Z [F.«(y.t)],-Q I $%Wlm-jv{y,t)]t=0dy.
J s = 0 s=0
It is easily verified that the operators
2m-l 2m-l
xs l^ ^jsiy 2m-j ^ss iy 2m-s * /-i ^jsxv2m-j
j=s j=s+l
= (-i)ip|-s+i^s«(o.o o.2.)(*.o)i?t2"-+i +2m£nN2m_j
j=s+l
make up a Dirichlet system of order 2m with infinitely differentiable
coefficients on dla+.
If we denote by Wj the F's's for which Fs = Sfs and by ^ the F's's
for which Fs = 38 j (in (2.16)), we obtain the following Green's formula
(2.17) JV
w) u ^y <Z* - j u j/* v dy dt
m— 1 m— 1
= £ J PjHVjvdy-Z f a.uFJZdy
J=°d;o+ J=°dj,+ J J
and, according to the very construction of the -Fj's, we verify that the
order of ^ is 2m — 1 — juj and the order of ^j is 2m — 1 — ntj.
Following the remarks of 1), formula (2.17) proves the theorem. D
Remark 2.1. From. (2.3), we easily deduce
Corollary 2.1. If u e 2(D), then Bsu = 0, / = 0,. . .,m — \,if and
only if
(Au) v dx = u A* v dx
Vv e 2(D) verifying Cs v = 0, / = 0, . . ., m — 1.
The necessary condition is obvious; conversely, if the condition
holds, we have, according to (2.3):
"-1 r
£ \ Bj u Tj v da = 0
j=o J
Vv e 2(D) verifying CjV = 0, / = 0,...,m— 1. Then, it is sufficient
to take v such that we also have Tj v = Bj u, j = 0, . . ., m — 1, which
is possible according to Lemma 2.2. From which we obtain BjU = 0,
/ = 0, . . ., m — 1. D

120 2. Green's Formula and Adjoint Boundary Value Problems
Remark 2.2. Formula (2.3) may be extended by continuity to the
case u, v eH2m(Q). Indeed, 3f(Q) is dense in H2m{Q) (see Chapter 1,
Theorem 8.2) and the integrals on r in (2.3) are well-defined thanks to
the trace theorem for the elements u of H2m(Q) (see Chapter 1,
Theorem 8.3).
2.4 A Variant of Green's Formula
With the elliptic operator A still defined by (1.5) with (1.6), we may
associate with it the sesquilinear form given by
(2 18) a(utv)=( £ apq{x) Dp uDqvdx.
v ' ' £ |pU«I*»
Then, if {F^f^o is a Dirichlet system of order m} with infinitely
differentiable coefficients on r, we may define a system {0j}fJo which
is normal on r, has infinitely differentiable coefficients, with: order of
Fj 4- order of 0j = 2m 4- 1, such that
m-l
(2.19) a(u,v) = f (A u)vdx - £ [OjuFjvda yiu1ve^{D).
q J ur
Indeed, as in Section 2.3, via "local maps", we may consider the
case of the half-ball a+ and then, in the same notation and with the
same reasoning as in Section 2.3, 1) and 3), we have, for u, v e
e@(o+ u d1a+) and vanishing in a neighborhood of d2o+:
\{^u)vdydt= £ \apqDpul)*vdydt +
m-l
+ Z I [Nn-juD[*]tm0dy,
J=0 aj+
where the system {Nm_J}jl=o is normal on d^a+ and has infinitely
differentiable coefficients, the order of Nj being 2m — j — 1; from
which, by reasoning analogous to the case of Section 2.3, 3), we deduce
(2.19). D
Remark 2.3. As in Remark 2.2, there is no difficulty in extending
(2.19) to functions ut v e H2m(Q). D
Remark 2.4. If A = A* (A is formally self-adjoint), then a(u,v)
= a(v,u) and, writing (2.19) for the couples ut v and v, u, we deduce
from it that
(2.20) \u~vdx- j (A u) • v dx
m-l m-l
= Z \FjU 0jVdo - Y \@jUFjvdo.
J=o £ j=o<

3.1 Two Lemmas
121
(2.21)
2.5 Formal Adjoint Problems with Respect to Green's Formula
We again consider Green's formula (2.3) and introduce
Definition 2.2. In the notation of Theorem 2.1, the system {C,}™^1 is
said to be the adjoint of the system {Bj^Jq with respect to A and to Green's
formula (2.3).
Also, the boundary value problem {^4*, C}
A* u = f in Q
Cju = gj on T, j = 0, . .., m — 1
is called the formal adjoint problem of (1.8) with respect to Green's
formula (2.3). D
Remark 2.5. Since the system {Cj}?=o depends on the choice of the
system {S^Jo, there exists an infinity of formal adjoint problems
of (1.8). However, it is easy to see from Remark 2.1 that all adjoint
systems of {£,-} are equivalent, in the sense that if {Cj} and {Cj} are
two such systems and if ue@(D), the conditions CjU = 0, / = 0,...,
m — 1, imply CjU = 0, / = 0, . . ., m — 1, and conversely. D
Now, assume that the operator A is properly elliptic and that the
system {Bj}fJo covers A . Then, it is natural to ask whether the formal
adjoint systems of {B^JJq, in Definition 2.2, cover the formal adjoint
A* of A. The answer is yes; in fact we have
Theorem 2.2. If A, as defined by (1.5) with (1.6), is properly elliptic,
the system {B^fJo, normal on r, covers A if and only if each normal
system {C/}™Jo\ cidjoint of the system {BjYjZo with respect to A and to
Green's formula (2.3), covers A*.
It is possible to give a direct, purely algebraic proof of this theorem
(see Schechter [2], Appendix II); however, it will be more convenient
for us to obtain this result by an indirect method in Section 4.3, below. D
Remark 2.6. It is easy to see, for example using (2.19) for A and A*,
with Fj = yJf that the Dirichlet problem {/4,y} for A (i.e. Bj = y3)
admits the Dirichlet problem {^4*, y) for A* as a formal adjoint problem.
3. The Regularity of Solutions of Elliptic Equations
in the Interior of O
3.1 Two Lemmas
The following lemmas will be used in the sequel. We recall that, if K
is a compact subset of Rn, we denote by HSK (Rn), arbitrary real s, the
subspace of Hs (Rn) of distributions with support in K. For every e > 0,

122 3. The Regularity of Solutions of Elliptic Equations
the injection of HsK(Rn) into Hs£e(Rn) is compact; this is an immediate
consequence of Theorem 16.1 of Chapter 1.
Lemma 3.1. For every positive real s and e > 0, the norm of the
injection of HSK (Rn) into Hs£e (Rn) tends to zero as the diameter of K tends to zero.
Proof. We recall that the norm under consideration is defined by
(3.1) N(K)= sup H|
Hs-e(Rn).
lljJs(Rn)-1
If the lemma was not true, there would exist a sequence of functions
^e#s(Rn) with the properties: support of ut contained in a ball of
fixed center x0f with radius 1/i, ||M{||h*(R") = 1 and || ut ||h»-«(r») does
not tend to zero. Then, according to the compactness of the injection of
HsK(Rn) into HK~e(Rn), we may assume that there exists a subsequence,
still denoted by ut) such that ut -► u =)= 0 in Hs~e(Rn). Since u
necessarily has support {x0} and s > 0 and since we may assume s — e ^ 0,
we obtain u = 0, which is absurd. D
Lemma 3.2. Let s be an arbitrary real number, I a positive integer
and let a be a C00 function in R", bounded together with all its derivatives.
Let o(q) denote the ball in Rn centered at the origin and of radius q] we
assume that 0 < q ^ £0/2, q0 > 0 given. Then Vw e Hsaie)(Rn) and Vp
such that \p\ = /, we have
\\aDpu\\Hs-HRn) ^ c(max |a\) \\u\\HsiRn) + LQ \\u\\Hs-HKn)y
where c = constant and LQ > 0 depends on q.
Proof. Let cp0 be a function in @(Rn), 0 ^ <p0 ^ 1, <p0 = 1 on g(q0) .
For every u e HsGiQ) (Rn), we have
a Dp u = <p0 a Dp u = Dp (<p0 au) + B u,
where B = linear differential operator of order </, with coefficients
in @(Rn) with support in cf{Q0).
Therefore, applying Lemma 7.2 of Chapter 1, we have
(3.2) \\aDpu\\Hs-HRn) ^ c1 \\<p0au\\HsiRn) + c2 \\u\\Hs-HRn).
We now introduce the functions cpQ e 2 (Rn) such that 0 :g cpQ ^ 1,
cpQ = 1 in <y(Q) and with support in g{2q).
We apply Lemma 7.1 with <p = (pe(p0 a; we obtain
(3.3) || <p0 a u\\HHRn) = || <pe <p0 a u ||H,(Rn) ^
^c(max\<pe<p0a\) \\u\\HHRn) + cc || w||H,-1(Rn) ^
~ C(m2a^'a|) I|W||HS^> + Ce IMlH-i(Rn)>
with cQ depending on cp0t cpQ) a and /.

3.2 A priori Estimates in Rn
123
The lemma follows from (3.2) and (3.3). D
We note that if a(0) =0 and s > 0, we obtain, using Lemma 3.1,
the existence of a function s(q) > 0 such that
lim e(g) = 0
and such that e
\\aD>u\\H.-l(Rn) ^ e(Q) \\u\\mRn), VueHsaie)(R»), \p\ = I. Q
3.2 A priori Estimates in RR
Let
(3.4) A=A(x9D)= £ ap(x)D>
be a linear differential operator of order I with infinitely differentiable
coefficients which, together with each of their derivatives, are bounded in R".
Denote by
A0 = A0(x,D) = £ «,(*)#*
\P\ = i
the homogeneous part of degree / of A. Assume that
(3.5) the operator Ao(0,D) with constant coefficients is elliptic.
Then, we have
Theorem 3.1. // A is defined by (3.4) and satisfies (3.5), for every
integer r ^ 0 there exists a positive number q0 such that if Q < Qo and
if u eL2(Rn) vanishes outside the ball a(g) with center at the origin and
radius q, and A u e H~l+r (Rn), then ueHr(Rn) and
(3-6) IMIhw) ^ Cr,e{\\A u\\H-l+riRn) + |M|Hr-1(Rn)}
(CrtQ depending on r and q, but not on u).
Proof. 1) Here, as well as in the sequel, c, C, Cr, CrQ shall denote
positive constants which may change from one inequality to another.
We first prove the theorem for r = 0. From (3.5), we deduce that
1 + |f|2^c|^0(0,f)|2 + 1 VfeR",
therefore also
M0(o,f)|2 l + |f|2a_1)
1 <c ' ov '\ + ' ' 2. VfeR".
" l + |f|2i 1 + lfl21
Multiplying by |^(f)|2 (u denoting the Fourier transform of u with
respect to xlt . . ., xn) and integrating on Rn, we obtain
(3.7) IMIi»aM ^ C{Mo(0,D)«||H-I(RB) + ||«||h-.(r.,}.
But, thanks to the hypotheses on the coefficients of A(x,D), we
may apply Lemma 3.2; therefore, there exists a £0 > 0 such that, for u

124 3. The Regularity of Solutions of Elliptic Equations
with support in <r(g) and q < q0> we have
(3.8) \\Ao(0,D)u-A(x,D)u\\H-l(Rn}£
^ \\Ao{0,D)u-Ao{x9D)u\\H-tiRn} +
+ \\A0{x,D)u - A(x,D)u\\H-liRn) ^
1
2~c
^T7llwllw(R") + CQ II^Hh-krh)
and therefore from (3.7) and (3.8) we deduce
(3.9)
IMlL2(Rn) ^ C\
A(XtD)u\\H-nRn) + — \\u\\L2<Rn) + Ce ||«||H-l(Rn)J
from which we obtain (3.6), with CQ depending on q.
2) Now, using the well-known method of "differential quotients",
we shall prove the theorem for r > 0.
Indeed, since the theorem is valid for r = 0, we may proceed by
induction. Thus, we assume that u e Hr~1(Rn)f r ^ 1 and that (3.6)
holds with r — 1 replacing r.
For h 4= 0, we set
u(xl9...txi_ltxi + htxi+lt...txH) -u(xlt...txH)
(3.10) quh u (x) = ,
h
i = 1, . . ., n.
If h is sufficiently small, the support of qith u is contained in the
ball o(q') with radius
Q' = Q + (go + g)/2 < £0>
if the support of «(#) is in <r(g). Thus, we may apply (3.3) with r — 1
to Qith u and we obtain
(3-11) \\Qi,hU\\Hr-HRn)^
^ Cr.Q'{\\A(XtD) QiihU\\H-l+r-liRn) + ||ei.*«||Hr-2(Rn)}.
We set Tiffc u = u(xlt . . ,txi_1,xi + htxi+1, . . .,xn) and we verify
that
^ (*, D) eiifc « - Qith A(x,D)u= £ [Qt.h «PM] #pTM «
and therefore
\\A{xiD)qithu - Qi,hA(x>D)u\\H-l+r-HRn) ^ Cr J|w||Hr-1(Rri).

3.2 The Regularity in the Interior of D 125
Then, from (3.11) we deduce
(3.12) \\Qi,hu\\Hr-HRn) ^ Crte{\\QihAu\\H.l+r.HRn) +
+ \\Qi,hU\\Hr-2iRn) + ||«||nr-l(Rn)} ^
^ Cr.Q{\\A «||fl-l+r(Rn) + ||«||flr-l(Rn)}
from which we deduce the fact that qih u remains in a bounded set of
Hr~l (Rn) as h varies; but then we can find a sequence h -* 0 such that
du
Qihu ~* Xi m #r_1(Rn) weakly; but qihu-* in ^'(Rn), so that
du ' dxi
-— e Hr~x (R") for i = 1, . . ., n, therefore ueHr (Rn) and we also
have (3.6). D
3.2 The Regularity in the Interior of Q and the Hypoellipticity
of Elliptic Operators
From Theorem 3.1, we shall now deduce the regularity of solutions
of elliptic equations in the interior of Q.
Let Q be an arbitrary open set in Rn and let
(3.13) A=A(x,D)=% ap(x)D>
be a linear operator of order / with infinitely differ entiahle coefficients
in Q and elliptic in Q.
We denote by H[oc (Q), arbitrary interger r, the space of distributions
u on Q such that cp u e Hr (Q) for all cp e 2 (Q). We have
Theorem 3.2. Let A be defined by (3.13) and r be an arbitrary integer;
if u is a distribution on Q such that A u e H^Q[+r{Q), then u e Hrloc(Q);
in particular, if A u is infinitely differentiable in Q, then u is also infinitely
differ entiahle in Q.
Proof. 1) We first show that
(3.14) if u e H\;cl (Q) and Aue H^r (Q), then u e H\oc (Q).
Let cp e2(Q) be arbitrary and fixed; set v = cp u and denote by v
the extension of v by zero outside of Q.
Of course, veHr-1 (Q) and veHr~l (Rn).
Also, A v e#ioc+r(i2), since
A v = cp A u -f Axu>
where A1 u is an operator of order / — 1 and therefore Axue H{~0[+r(Q)
since, by hypothesis, u e H\~,} (£}).
Now assume that r — 1 ^ 0. Because of the local character of the
theorem, it is sufficient to show that if cp has its support in a ball a (q)

126 3. The Regularity of Solutions of Elliptic Equations
with center at the origin and with sufficiently small radius q, then
v e Hr(Rn). But then we may apply Theorem 3.1, since r — 1 ^ 0 (of
course, we may assume to have extended the coefficients of A to R"
so that Theorem 3.1 applies); which shows that vEHr(Rn) and
therefore that (3.14) is valid.
Now, let r — 1 < 0. Then, we may write v in the form
v = (I - A)k w, with k such that r - 1 + 2k ^ 1.
In fact, it is sufficient to set
/ 1 \\
$(£) = { v (£) {v (£) = Fourier transform of v)
and to take w = Fourier anti-transform of $.
Then w e Hr~1 + 2k(Rn) and its restriction w to Q belongs to
Hr~l + 2k(Q). FinaUy, we have
v = (I — A)k w.
But A (I - A)k w eH{-Q[+r{Q) by hypothesis and the operator A (I- A)k
is elliptic and of order / + 2k.
Therefore, we may apply (3.14), for the case already demonstrated,
to w\ thus, w e H[^k{Q) and therefore v e Hrloc(Q) and (3.14) is proved
for arbitrary r.
2) If u is an arbitrary distribution on Q and A u e H{^lc+r(Q)t we
note that, because of the fact that every distribution is locally of finite
order, for every ball co such that a> a Qf there exists an integer r0
such that ueH\°oc((d).
So that if r0 ^ rf the theorem is obvious; if r0 = r — 1, we may
use (3.14). Finally, if r0 < r — 1, we have
ueH\Ic{cd)> A UEH-lc+r{a>) cz #roc+ro+1 M
and therefore, thanks to (3.14), ueH\^1 (co) . Now, we may proceed in
the same way a finite number of times until we have shown u e H\oz (co)
and therefore the theorem is proved. D
Remark 3.1. Following a terminology which by now has become
standard, we say that a linear differential operator A with infinitely
differentiable coefficients in Q is hypoelliptic if u e & (Q), A u eC°{Q)
implies u e C°° (Q).
Thus, we have shown that if A is elliptic, than it is hypoelliptic.
Furthermore, we shall see in Volume 3 that if the coefficients of A
and / are analytic (or belong to a Gevrey class) in Q, then u is analytic
(or belongs to a Gevrey class) in Q.

4.1 A new Formulation of the Covering Condition 127
Remark 3.2. In the preceding theory, we may also use the spaces Hr
with arbitrary real r\ in particular, Theorem 3.2 is valid for all real r
(see, for example, Schwartz [7]). But, we shall only use the case integer r
in the sequel.
4. A priori Estimates in the Half-Space
4.1 A new Formulation of the Covering Condition
The "'a priori" estimates on the boundary shall be given first in the
case of the half-space and for operators with constant coefficients.
We shall use the notation already introduced in Section 2 for the
point x e R": x = (y, t), with y = (ylf . . ., yB-1) e R""1 and t e R. The
"dual" variable of x will be denoted by f and represented by f = (rj, t'),
with tj = (rj1, . . ., rjn_ J e R"~* and feR.
R+ denotes the half-space of R" of x's with t > 0.
Let
{A(D)u = A(Dy,Dt)= £ apDpy'D!»,
(4.1) W = 2m
I P = (P'>Pn)> P' = U>l,...,Pn-l)
be a differential operator with constant coefficients, homogeneous of
degree 2m; let
(Bj(D)=Bj(D„Dt)= £ ft^Dj'^,
(4.2) |ft|=m'
I h = (h',hn), A'= (*!,... A-i), / = 0,...,ro- 1
be m differential operators with constant coefficients, By being
homogeneous of degree mj} with
(4.3) 0 ^mj ^2m - I.
We assume that A is properly elliptic, that is, thanks to Remark 1.2,
we assume that
I) the characteristic form A(tj) = A(tjt t') is different from zero for
all f 4= 0, f e R"; *md /or all rj e R""1 *md 4= 0, the polynomial A(rj}r)
in the complex variable r has m roots with positive imaginary part.
We denote by t* (rj) (resp. rj~ {rj)) the roots of A (rj, r) with positive
imaginary part (resp. negative imaginary part) and set
m m
(4.4) M* fo, t) =* IJ (* - *,* fo)) = E c** fo) *""*•
i=l fc = 0
We have
M+(tj,t) = (-l)mM-(-7j, -r)

128 4. A priori Estimates in the Half-Space
and the coefficients c£ (rj) are analytic functions of rj e R"~* and rj =)= 0,
and homogeneous of degree ft.
For every / = 0, . . ., m — 1, we also consider the polynomial in r
fc = 0
Then, we have
Proposition 4.1. For tjeR""1 and =)= 0, we have
(4.5)
—-j-iW.t) rkd , 0^i<m-l, 0<k<m-l,
2ni) M+(t],r) Jk ~'~ ~ ~
V
for every rectifiable Jordan curve y in the complex plane, which encircles
all the roots x^ (rj).
Proof. We not that, if / ^ ft, M^-j-i rk is a polynomial in r of
degree m — 1 — / 4- ft for which the term of maximum degree is given
by a$ (rj) rm~1~J+k, In that case, it suffices to deform y into a circle with
center at the origin and to make the radius of the circle tend to infinity
to obtain (4.5). If / < ft, then Mm_J_l rk = t*-3~1 M+ + Q, where Q
is a polynomial of degree < ft — 1 and, therefore, to obtain (4.5), it
f Q
suffices to calculate —— dr, and this integral vanishes because the
J M+
v
degree of Q ^ m — 1 (thus, we may again deform y into a circle of
radius tending towards infinity). D
We shall also make the hypothesis that the system {Bj}™^ covers A,
that is:
II) For all TjeR""1 and =)= 0, the polynomials Bj(rj,r) in r are
linearly independent modulo M+ (tjtr).
Therefore, if we set
(4.6) B] (r,, T) = Bj fa, r), modM+ (r,, r), (i.e. Bs = QjM+ + B's)
hypothesis II) is equivalent to the fact that, if Bj(rjfr) is given by
m-l
k = 0
I the determinant of the matrix \\ b'Jk (rj) II is =)= 0
forrj eR""1 and +0.
We shall also require another equivalent formulation of hypothesis II).
For this purpose, we consider, for all rj eR""1 and =)= 0, the ordinary

4.1 A new Formulation of the Covering Condition 129
differential equation
(4-8) A^-L^^-O
with the conditions
= cJ} j = 0, . . ., m — 1,
(4-9) b'(*t4)*(m
t = 0
where the c/s are given arbitrary complex numbers. In agreement with
the usual notation (see L. Schwartz [1]), we denote by ^(R+) the space
of functions <p, infinitely differentiable for t ^ 0 and rapidly decreasing
for t -* +oo (i.e. tk<pU)(t) -* 0 as t -* +oo, Vft, V/).
We shall no prove
Proposition 4.2. Assume that hypothesis I) holds, then hypothesis II)
is equivalent to one of the two following conditions:
IF) problem (4.8) —(4.9) admits a solution belonging to £f(R+), for
all cJf j = 0, . . ., m — 1;
II") problem (4.8) —(4.9), with Cj = 0, admits only the null solution
in ^(R+).
Proof. 1) Let us first show that IF) is equivalent to II"). It is a
well-known fact that the solutions of (4.8) are in the form of exponential-
polynomials; therefore, we deduce from I) that the space of solutions
of (4.8) which belong to ^(R+) has dimension m. Thus, we may
construct a mapping of this space into Cm by making correspond to each
solution cp of (4.8) which belongs to^(R+), the vector in Cm having
1 d \ |
, 7 = 0, . . ., w — 1. This mapping is
t = o
a linear mapping of an m-dimensional vector space into Cm; therefore,
it is surjective (i.e. IF) is satisfied) if and only if it is injective (i.e. if
II") is satisfied).
2) Now, we show that II) is equivalent to IF).
Assume that II) is satisfied and let fy, / = 0, 1, . . ., m — 1, be m
given complex numbers and r\ e Rw_1, #= 0; then, because of (4.7), the
system
m— 1
(4.10) lib'js(rj)qs(rj)=cJ, / = 0, ...,m-\
s = 0
admits a unique solution {qs{r])}T=o which depends on tj. We set
(4.11) u(riJ)=-—\ I?s(?7) M+l 7—e Tdr,
2n\ J s=o M+ (?j, t)
Y
where y is a rectifiable Jordan curve which encircles the roots r+J(?j)
of M+ (rj,r). We may assume that y is in the complex half-plane cor-
components Bj [ tj, <p

130 4. A priori Estimates in the Half-Space
responding to positive imaginary parts, since the t*(?j)'s are in this
half-plane. Then, the function t -* u(rj,t) e^(R+) and we easily verify
that it satisfies equation (4.8); furthermore, for t = 0, we have
\ i it) 2niJ s=o M+(rj,r)
v
1 f*m— 1 m— 1
zjr l J s=o k = o
fo)-
M+fo,r)
y
m-l
= (because of (4.5)) = £ 6J, fa) &(??) = cJ}
s = 0
j = 0, . . ., m — 1
and therefore w satisfies II').
Conversely, we assume that II) does not hold; this is equivalent to
the fact that the determinant of || bfJk (rj) || vanishes and therefore there
exist non-null solutions {qs(ri)}?Jo of system (4.10) with
cj = Of / = 0,.. .,m - 1.
We show that u, as given by (4.11), is not identical to zero. Indeed,
there exists s0, 0 5* s0 ^ m — 1, such that qSo (rj) 4= 0 and therefore
dSou{r),t)
df°
y
= (because of (4.5)) = c5° qSo(rj) 4= 0;
so that u is not identically zero. But, this contradicts II") and
therefore II'). D
4.2 A Lemma on Ordinary Differential Equations
The following lemma is important for the sequel.
Lemma 4.1. Under the hypotheses I) and II) of Section 4.1, for all
TyeR""1 and 4= 0, the operator
is an (algebraic and topological) isomorphism of H2m (R+) onto L2 (R+) x Cm.
Proof. Using the Banach theorem, it is sufficient to show that 9n
is an algebraic isomorphism; thanks to Proposition 4.2 and the fact
that the solutions of (4.8) in H2m(R+) (and even in «5^'(R+)) are in

4.2 A Lemma on Ordinary Differential Equations
131
y(R+), it suffices to show that 0>n is surjective, i.e. that there exists
<peH2m(R+), solution of
(4.12)
A[rj,j—\<p(t) =/(<) t>0,
*'(*l4)'W
= cJf j = 0, . . ., m - 1,
t = o
where arbitrary / eZ,2(R+) is given and the c/s are arbitrary complex
numbers.
Let / be the extension of / by zero for all t < 0 (i.e. f(t) = f(t) if
t ^ 0 and f(t) = 0 if t < 0). There exists y(*) e#2m(R), solution of
(4.13) ^^IAJ^)=/W.
By Fourier transform in tt it suffices to take
/>)
W) =
^fo,o
(?) = Fourier transform of v)
and to note that A (rj, t') #= 0 for real t'; then the Fourier anti-transform
(t'V
w oi w belongs to H2m(R), for —— is bounded if t' e R for / < 2m.
^ A(rj,t')
Furthermore, y) satisfies (4.13).
Consider the restriction \p1 of y) to R+; \p1 e #2m(R+) and therefore
we may calculate
s,e?4^>iW
= Yj> / = 0, . . .,m - 1.
t =o
Now we apply Proposition 4.2 to solve the problem
.4 (,,}£) „,,(<>-<>, (>0
M't^H
cj - 7j> j = 0,. . .,m - \
in S?(R+).
Then y2 eff2m(R+) and q>(t) = \px (t) + y>2 (t) solves problem (4.12). D
Remark 4.1. Setting Cn = || ^1 ||^(L2(R+)xc-;H2m(R+)), we deduce
from Lemma 4.1 that
(4-14) IM*.^t,£CJ^||uau)x(> Vy6^2-(R+).
We shall see that the function ?j -► Cv is continuous in Rn~* — {0}.

132
4. A priori Estimates in the Half-Space
Let
G = &(L2(R+)xCm;H2m(R+));
the function rj -* || 2P~l ||G is bounded on every compact set of
Rnl - {0}.
Then, for rj, rj0 e Rn"x - {0}:
yields, as r\ -* rj0:
W&V - 9* Hg ^ c II &, - &no IIf- -F = ^(^2m(R+) J ^2(R+) x C"),
and \\^n — ^no ||F -► 0 as rj -► ?^0 (for ^4 and the B/s are polynomials
in rj). D
Remark 4.2. In order to point out the role played in the theory of
this chapter by the hypothesis that A be properly elliptic and by the
hypothesis that {Bj^Jo1 covers A (see also Section 8.4, further on), we
note that the hypothesis on the roots of A (rj, r) in I) and hypothesis II)
are also necessary for the validity of Lemma 4.1.
Indeed, assume first that I) is satisfied, but that II) is not satisfied
for some rj 4= 0; then, as we have seen in the proof of Proposition 4.2,
there exists a u e£f(R+), which satisfies (4.8) and (4.10) with Cj = 0,
/ = 0, . . ., m — 1 and which is not identically zero; which contradicts
the lemma.
Next, if we assume A to be elliptic, but not properly elliptic, then
we may assume that, for some rj 4= 0, there are m' > m roots rt (rj)
m'
of A(rj,r). We set M+ (rjfr) = Y\ (r ~ xt (*?)) and> because m < m',
there exists a function u (t) eSf (R+) and 4= 0 which satisfies the equation
M+ ir], J u(t) = 0, t > 0, and consequently also (4.8) and the
\ i dt ]
boundary conditions (4.10) with cj = 0, q.e.d. D
Further on, we shall make use of inequality (4.14) to obtain a priori
estimates in the spaces Hs(Rn+) with s ^ 2m. But, we shall also need
certain "dual" estimates; for this reason we shall now study the adjoint
mapping 9% of SPn.
&>* is a continuous linear mapping of L2(R+) x Cm into (#2w(R+))',
defined by the formula
(4.15)
(<p,P*Yy = <^„9>, »*>, V(peH2m(R+) and ^eI2(R+) x Cm,
where W denotes the element {ip; c0, . . ., cm_l} of L2(R+) x Cm and
the brackets denote the appropriate dualities.
But (H2m(R+))' may be identified (see Chapter 1, Section 12.6)
with H^2m(R). Then, we denote by cp0 a fixed extension of the functions cp

4.3 First Application: Proof of Theorem 2.2 133
of H2m(R+) to H2m(R) (such an extension exists and depends linearly
and continuously on cp) see Chapter 1, Section 2.2), ip the extension of
ipeL2(R+) by zero for t < 0, d the Dirac measure at the origin; we
may write (4.15) in the form
(4.16) <y,^*y>
f = 0
((D)
where A* and B* are the formal ad joints of A and Bj (as differential
operators in t) and the brackets denote the appropriate dualities. Thus,
0** may be defined by
/ 1 d \ m~1 I 1 d \
(4.17) K'-"{i.1-S)*»+£l*:(<i.T-i;)',>-
From Lemma 4.1, we deduce the estimate
(4.18) ||^||L2(R+)xC^ ^ C, \\0* !P||h-2-kr, V^eL2(R+) x C-.
4.3 First Application: Proof of Theorem 2.2
Before going on with the study of a priori estimates in the half-space,
we shall give an application of Lemma 4.1 by proving Theorem 2.2.
Let A and {BjYjZo satisfy the hypotheses of Theorem 2.2. To prove
the theorem, it is obviously sufficient to prove that the system {Cj}™^
covers A*, where {Cj^Iq is given by Green's formula (2.3).
Via "local maps" and "partition of unity", as in Section 2.3, 1)
(the notion of covering having a local character), we are brought back
to the case of the half-ball o+ and to Green's formula (2.17), using the
notation of Section 2.3.
Therefore, for every u and v belonging to @(Rn+) and having support
in o+ u d1o+, we have
(4.19) f (s/u)vdydt- (ustf*vdy
dt
R" R +
-1
= 1 \ yjuVjvdy-Z f ajufjvdy
'=°r»-i ;=°r»
((D) We use the fact that c d € H-2m+mJ (R) and that if v € H2m~mJ (R), then
<i;, c (5) = (v (0), c), the first bracket denoting the duality between H2m~mJ (R)
and H~2m+mJ (R) and the second bracket denoting the product v (0) c.

134
4. A priori Estimates in the Half-Space
and the problem is to show that {^J^o1 covers j/*, under the
hypothesis that {^j}7=o covers s/.
Let A be a real number, X > 1: then the functions u(Xy,Xt) and
v(X y, X t) are still in Q){R\) and have support in <y+ u d1a+; therefore,
we may apply (4.19) to u(Xy, Xt) and v(Xyt Xt) and, after some easy
calculations, we find, in the obvious notation,
Z2 \ap\j,^(D>u{y9t))v{y,t)dydt-
-"i f I z *»
j = 0 J l|ft| = ^
Rn-1
j)o*«(y.<)
f = 0
|*|=2iii-1-^ \A/
Rn-1
dy +
*=o
x Z tJh[^-)D''v(y,t)
|ft| = 2w-l -mj \ A>
*-°(t)-
Then, letting A -► +00, we obtain
(4.20) f (^0(0, D) u{y, t))v(y, t) dy dt - [ u{y,t)^fv{y~i) dy dt =
= "l f ^.o(0.D)«(y,<)
i-o R„J-,
W-l
-X f ^iO(0,D)«(y,<)
«W0,-D)»(y,<)
^y —
t = o
rj.o{0,D)v(y,t)
t = o
dy,
where j/0 an^ J&o, &j,o> ^j.o* ^j,o> ^j,o nave constant coefficients
and are homogeneous of suitable degree: "by homogeneity", we obtain
without difficulty that (4.20) is also valid for arbitrary u and v in £^(R+).
Now, let q>(t) and \p(t) be given functions in ^(R+), %(y) be given in
^(Rn_1) and real X > 0; we apply (4.20) to the functions
and
v{y,t) = e^>X[Xy)xp{y)

4.3 First Application: Proof of Theorem 2.2
for all r\ eR""1 and 4= 0. It follows (write
s/o{0,D) = ^0(0, £,,£,),...)
135
that
+ 00
U0{^^r},—^<p(t)\w{t)dt
{<p(t)*?z(o,iv>^Jv(t)dt\ J \x(y)\2dy-
0 Ir»-i
-{^"(•■"■^I'W
f = 0
O.o(0,ii?, —JvW
m-1 / i \ / i \
- Z ^.o(o,in, —\<p(t) fj.ofo,i»?,—Jy(0
x / lz(y)l2rfy = o(A).
Rn-l
Letting A -» 0, we obtain
+ 00
+ 00
J Uo\0Jv.Jj)f(Ay>{t)dt-\ <p(t)s/*(o,iri,-j-\y>(t)dt
(4.2i) = m£y;,0(o, iif,-^ ?w I Vj.0(o, iv,-^ f{t)
m~l ( d\
f = 0
_ ^j.o[0,iri, — )f(t)
f = 0
for all p,^6^(R+) and therefore also for all <p, ip e H2m(R+), since
^(R+) is dense in #2m(R+).
At this point, to show that ^SjYjZo covers *s/*, it is sufficient (for
conditions II") and II) of Section 4.1 are equivalent (Proposition 4.2))
to show that the differential problem
(4.22)
jf*(o,iri,— jf{t) = 0, t>0,
= 0, / = 0,...,m - 1,
0.o(0,i»/, — )y(/)
f = 0

136 4. A priori Estimates in the Half-Space
admits only the solution y)(f) = 0 in H2m(R+) (and therefore also in
«^(R+)). Therefore, let ipeH2m(R+) be a solution of (4.22). We may
apply (4.21) with arbitrary q>eH2m(R+) and obtain
(4.23)
+ 00
0
V.1 / d \
•^.o|0.ii^)vW
f = 0
But the system {Bj}f=o covers s/ and therefore s/0 and {&Jt0}
satisfy hypotheses I) and II) of Section 4.1; therefore, we may apply
Lemma 4.1, which shows that the problem
^o(o,i^—L$ = /M, t>o,
9j.o[0,iri,— )ip(t)
= 0, / = 0,. . ., m — 1,
f = 0
admits, for every / e L2 (R+), a unique solution in H2m (R+). Therefore,
from (4.23), we deduce that
+ oo
j hpdt = 0 V/eI2(R+)
o
and therefore y; = 0. D
4.4 A priori Estimates in the Half-Space for the Case
of Constant Coefficients
Now, let us again consider the operators A and {B/JJpJo1 given
by (4.1) and (4.2) under the hypotheses of Section 4.1 [i.e. (4.3), I)
and II)]; we consider the operator
(4.24) 9\ u -> 9 u = {A (D) u, B0(D) u |f=0, ...9Bm_t(D)u |f,0},
which according to the trace theorem of Chapter 1, Section 8 is a
continuous linear mapping of H2m(Rn+) into
m-l
L2(Rn+)x Y\H2m-mJ-ll2{Wl).
J = 0
We denote by ^* the adjoint of 9 which maps
m-l
L2(Rn+)x [] ^_2m+mj + 1/2(Rn_1) in*o (H2m(Rn+))'.

4.4 Half-Space lor the Case of Constant Coefficients 137
Then, we proceed in the same way as we did to obtain (4.17). If
m-l
F = {flgo, - - •,gm-i}eL2(Rn+) x J] ff-*»+»i+i/*(R-i)
j = o
and
q>eH2»(Rn+),
we have
?* F, 0> = <F, 9 <py = </, il ?> + £ <g„ B, p |,=0>
m-l
where / is the extension of / by zero for t < 0, d (t) is the Dirac measure
at zero, q>0 is an extension of <p, <p -> 990 being a continuous linear
mapping of #2m(R^) into H2m(Rn).
Then
m-l
(4.25) ^* F = 4* (D) / + £ B* (D) (g, (y) ® 6 (t)),
j = o
where A* and £* are the formal adjoints of A and Bj.
Then, we have
Theorem 4.1. Under hypotheses I) #w^ II), /Atf following inequalities
hold:
(4.26) ||«||fl2m<Rn> ^ C{||^«||L2(Rn)x,5"1fl2«-«i-l/2(Rn-l) +
+ + j=0
+ II « 1H2-1(R-)} V«6^(R"+)
#w^, wz% ^* gw£» #y (4.25),
(4.27) ||-F||L2(Rn)xmn H-2m + my+l/2(Rn-l)
+ /=S°
m-l
C denoting a constant (independent of u and F).
Proof. 1) C shall denote various constants.
Again, we use the notation of Lemma 4.1 and the inequality (4.14)
for \rj\ = 1; Cv being a continuous function of r], we obtain, writing

138 4. A priori Estimates in the Half-Space
out the norms explicitly:
+ 00
IjVw^c m^.i-i)^)
dt +
m-l
+ E
J = o
BAv.~)<P«)
f = 0
V<pe#2m(R+), \rj\ = 1,
C being independent (of (p and) of rj.
Since the operators A and Bj are homogeneous of degree 2m and w^
respectively, we obtain, for arbitrary r\ 4= 0,
2m
E \v\2(2m-°
1 = 0
l9)(,)(<)|2rf^ C
^{^
dt +
m-l
+ y |„|2(2m-mj-l/2)
B,h.-J4)*0
C being independent of r\.
We apply this inequality with q>(t) replaced by u(rj,t) (Fourier
transform of u(y, t) with respect to y), with u e H2m(Rn+); then, setting
A(D)u(y,t) = f(y,t) and B,(D) «(y, /) |f=0 = gj(y) for the sake of
simplicity, we obtain
t a
m /*
E tol2<2—»
J=o J
S'dfo,*)
a*1
<*<
< c
t-oo
<)l!«+ I l1l2(!-"'-"2,llJ(1)l"
from which, integrating over \yj\ ^ 1:
+ 00
dlu(rj,t) |2
(4.28)
III
(1 + lfll
2\2m-l
< C
a*
J f \f(v,t)\2df]dt +
dt dr] ^
+
m-l /*
I (l + \fl\2)2m-m'-1,2\6Afl)\2

4.4 Half-space for the Case of Constant Coefficients 139
Now, consider the integrals over \rj\ ^ 1. We obviously have
(4.29)
2m-l /» p
.?. J J
(1 + \v\
2\2m-l
Ml* 0
dlu(r),t)
dtl
dtdrj
2m-i r r
SC,?o J J
dlu{rj,t)
dtl
dtdri^C H«||fl2m-i(Rn).
Furthermore, since A is elliptic, we may write it in the form
32m 2m-1 Ql
with oc0 #= 0 and the a^m-iWs being continuous functions of r\\
therefore, we have
d2mu(r,,t) ^ ..d'u(r]J)
1 = 0
8t2
8t'
whence
(4.30)
I
Ml* 0
d2mu{rj,t)
di
2m
dtdrj^C
T 00
i
\f{rj,t)\2 dtdrj +
Ml* 0
+ 00
+ S
M = i 0
dlu(r),t)
dtl
dtdrj
gC{||/||L2(Rn+)+ llWll^-I^n)}.
Then (4.26) follows from (4.28) and (4.30).
2) Now, we prove (4.27). In the notation of Lemma 4.1, we deduce
from (4.17) and (4.18) that
\y{t)\2dt +
+ CX
m-1 r
Z \cj\2 < C
j=o J
(1 + \t
'■' |2\-2m
\2mA*{rjtt') x
e-»'>(/)i/+ £im'£?(>M'K-
J = 0
rf/'
V?={^;c0,...,cm.1}eL2(R+)xCw and for [77I = 1,
where C is independent of rj.

140
4. A priori Estimates in the Half-Space
The operators A* and B* also being homogeneous, we obtain, for
arbitrary rj #= 0,
T 00
J I,
(t)\2dt+ £ i^i2^-2-*1/2)!^!2 ^ c
J=o
-p oo
J""
I + |*'|2)-2mx
i2mA*(ri,r)\ e-it'ty(t)dt+ £ imj B* (rj, *') Cj
<**'
We apply this inequality with ip(t) replaced by f{rj,t) and ci by
gj(rj), with feL2 (Rn+) and g; e Hmr 2m+ u* (Rn~*); setting
+ 00
1
v(rjjf) =
+ 00
1 f
\2n
e-i>'tf(V,t)dt=^= e"'<f{ri,t)dt,
sjln
we have
\f(v.t)\2dt+ Y,\v\2im'-2m+l'2)\tAv)\2
J = o
< C
+ 00
j (1,1
+ \n2r
i2mA*{v,t')v{rj,t') +
J = 0
it9
But the expression
w(n,t') =i2mA*(r)tt')v(r),t') + I im'£?(>?,*')£; fa)
j=o
is the Fourier transform in the variables (y, t) of ^* F for
f = {/;^o>...,^m-i};
thus integrating over \rj\ ^ 1, we obtain
+ 00
(4.31) f f \f(ri,t)2\dtdri+mJ:Q f (1 + |^|2)^2m+1/2 k.W I2 ^
+ 00
^C | f (1 + \V\2 + \t'\2)-2m\w(v,f)\2dt'drj
|„|ai o
^C||^*F||V»(R-).
Now, we consider the integrals over \rj\ g 1.

4.4 Half-Space for the Case of Constant Coefficients
141
First of all, since \rj\ ^ 1,
m-l
(4.32)
i
j=o
(i + \v\2)
2\mj-2m+l/2
\ij(v)\2drj
M*i
m-l (*
(1 +\V\
2\mj - 2m- 1/2
\gj{ri)\2dri
M^1
= ^ Z ll^jllHw,--2w-1/2(Rn).
By Fourier transform in tf, we immediately see that (— -f 7]
W* /
is an isomorphism of L2(R) onto H~2m(R)] therefore, we have
+ 00
\f(t]J)\2dt^c\
,2m
I
< c
d2mf(t],t)
dt2m
fo.O
12
12
\H-
\H-2m(R)
if-2n,(R)
+ II/0M)IIh-.<r>,
where, as always, f{rj,t) denotes the extension of f{rj,i) by zero for
t < 0. Hence
+ 00
(4.33) [ f |/(*M)|2^<fy
hi*1 o
r + 00
[\ri\£l -co
< C
"J
<?270?-')
a<2n
i»2
H-2™(R)
< c
(i +tol2 + l<T)~1l»fo.<')l2<*<'<ty +
hlgl -oo
a2"/fo.O
2m
< c
II/IIh-»(r») +
I
3*
MSI"
d2mhv,t)n2
drj
H-2™(R)
M£i
S*2
drj
if-2m(R)

142 4. A priori Estimates in the Half-Space
d2mf
In order to estimate the last term in (4.33), we express with
the aid of 0>* F; indeed, we may write ^4* (?;,/') in the form
2m-l
A*(rj>t')=K*t'f>+ £ <x*2m-,(V)t",
1=0
with <%* 4= 0 and <%*m -i{rj) continuous functions of rj. Therefore
2m-l
4(in2mHv>n=i2mA*(rj,t')v(r1>n- £ i2m<xtn.l(tl)f'v(v,f)
1 = 0
m-l
J = 0
2m-l
- £ i2"**»-i(i7)<"»fe,n.
from which we obtain
(4.34)
\n\ =
d2mf(f]J)
dt2m
drj
H-2m(R)
= I I (1 + \t'\2)-2m\{it')2mv(ritf)\2dt'dr)
\V\=1 -oo
(i + \n\2 + \v\2)-2m\w(n,v)\2Mdri +
-r oo
n
\v\=i -oo
+ 00
2m-l r f
•(I j I (i + \n\2+ \t'\2)-2mWHn,t')\2dfdn +
|*?|gl -co
+ 00 1
m-l /• /•
hl^ i -a
< C
J = 0
So that (4.27) follows from (4.31), (4.32), (4.33) and (4.34). D
4.5 A priori Estimates in the Half-Space for the Case
of Variable Coefficients
Let us now consider the case of operators with variable coefficients,
still in the half-space R+.

4.5 Half-Space for the Case of Variable Coefficients 143
Therefore, still with the notation x = (y,t) for the elements of R", let
(4.35) A(x,D)= X M*)£*
\p\Z2m
be a linear differential operator of order 2m with infinitely differ entiahle
coefficients which, together with all their derivatives, are bounded in Rn.
We consider A (x, D) as an operator in Rn+, but it is convenient to extend
the coefficients of A(x,D) to all of Rn.
We denote by
A0(x,D) = £ a9(x)D'u
\p\ = 2m
the homogeneous part of degree 2m of A(x,D).
Let
(4.36) Bj{y, D) = £ bJth(y) D\ j = 0,..., m - 1,
be differential operators of order mJ} with 0 ^ m, < 2m and with
infinitely differ entiahle coefficients which, together with all their derivatives,
are bounded in Rn_1.
We denote by
Bj.0{y.D)= X bJth{y)D\ / = 0,...,m-l,
the homogeneous part of degree mj of Bj.
We shall assume that the differential operators A0 (0, D) and BJt0 (0,D)
with constant coefficients satisfy hypotheses I) and II) of Section 4.1.
We denote by SP the operator
(4.37) ^: u-+&u = {A{x,D)u;B0(y,D)u\t=0,...,Bm_1(y,D)u\tm0}
which, thanks to the hypotheses on the coefficients of A and BJt is
a continuous linear mapping of
m-l
H2m(Rn+) into L2(Rn+)x Y\H2m'm^n[W1).
J=o
We denote by ^* the adjoint of 0>, which maps
m- 1
L2(R\) x [] #-2^+1/2(R»-i) ^to (tf2m(R"+))' = tf£„2m(Rn)
and which may be defined, as for the case of constant coefficients, by
m-l
(4.38) &* F = A*(x, D)f(x) + J B*(y, D) (gj(y) ® *(*))
m-l
VF = {/;g0,.. .,gm_t} eLHK) x nff2"-"i_,/2(R"-1).
J = 0
A* and £* being the formal adjoints of A and BJt and / still having
the same meaning.

144 4. A priori Estimates in the Half-Space
Theorem 4.3. Let A and {B/JJTo1 be defined by (4.35) and (4.36).
Assume that the operators Ao(0,D) and {Bj(0, D)}™^ with constant
coefficients satisfy hypotheses I) and II) of Section 4.1; then, for every fixed
integer r ^ 0, there exists a positive number q0 such that if q < q0 we have
i) if u e H2m(Rn+) and vanishes outside the ball g(q) with center at the
origin and radius q and
m-l
^wetf'(R„+)x f]#2m+'-mJ-1/2(R»-i)
then J=0
u e H2m+r (R"+)
and
(4.39) ||«IU«+,(R;,^Crfff
{ll^^llHr(Rn )X "n H2»n+r-»Fiy-l/2(Rn-l) + \\ U \\ H2m +r- 1 (Rn )}
+ /»0 +
(Cr,e depending on r and q);
m-l
ii) if F = {/;g0, . . .,6,,-J 6l2(R"+) x n^-2",+",i+1/2(»""1) ^
j = o
vanishes outside g(q) (i.e. / vanishes outside g(q) and gj(y) vanishes for
\y\ > q), then
(4.40) ||^||L2(Rn)xmn1H-2n, + n,y+l/2(Rn-l)
+ y-o
)m-l l
ll^*^llfl-2m(Rn) + ||/||fl-l(Rn) + £ II ?J llfl" 2m + my-l/2(Rn-1)| ,
J = 0 J
where C0tQ is independent of F.
Proof. 1) We prove i). First we verify (4.39) for r = 0.
If u e H2m (R+), we may apply Theorem 4.1 to the operators A0 (0, D)
and BjtO(0,D) and therefore we have
(4.41) \\u\\H2miRV ^ C JMo(0, D) u\\lhrv +
m-l |
+ X H£/.0(0» ^) «Hfl2m-my-l/2(Rn-l) + || « ||fl2m-l(Rn )} .
J = 0 + J
But, thanks to the hypotheses on the coefficients of A (x, D) and Bj(y, D),
there exists q0>0 such that, for u with support in or (q) , q < q0 , we have
(4.42) \\Ao(0,D)u-A(x,D)u\\LHRV
^ \\Ao(0, D) u - A0{x, D) u\\LHRV +
+ C'e\\u\\H2m.HR)^ X \\^p(0)-aP(x)]D"u\\LHRn) +
\P\ = 2m *
1
AC
+ C'e ||«||h2"-«(r») ^ 77; IMIhi«<r» > + C'e ||«||h2«-i(r»)

4.5 Half-Space for the Case of Variable Coefficients 145
and, using the trace theorem of Chapter 1, Section 8, also:
m-l
(4-43) X \\BJtO(0,D)u- BJf0(yfD)u\\H2m.mrl/HRn.i)
^— II «|| H 2m<R»> + C'Q ||«||fl2m-l(Rn).
Therefore, from (4.41) we deduce
HWllfl2*i(Rn) ^ C {\\ ^,^||L2(Rn)XWn H 2m - mj- 1/2 (Rn- 1) +
+ + y-o
+ Il«llfl2m-1(R»>} + \ ||«||fl2m<R»),
whence (4.39) with r = 0.
Now, we show (4.39) for r > 0 by induction on r. We know by
hypothesis that u e #2w(R+) and, as we have just seen, we also know
that (4.39) holds for r = 0.
Therefore, we assume that u e H2"1*'"1 (Rn+) and that (4.39) holds
with r replaced by r — 1, and use the method of "differential quotients".
For h #= 0, we set
Qi.h " (*) = ^ •
For sufficiently small h, the support of Qithu, i = 1, . . ., n — 1, is
in the ball cr(g') with radius @' = £ + (g0 — q)j2 < q0, if the support
of w is in (t(q). Therefore, applying (4.39) with r — 1, we have
(4.44) \\Qithu\\H2m+r-i(Rn+)^ CrtQ.{\\A(x,D) Qithu\\ar-iiRnj +
m-l
+ Z llBj(^»C) eMWllfl2m+r-mr3/2(Rn-l) + || Qi,h U II if 2m +r- 2 (R»,)} .
Set tm w = «(#!, . . ., xt_l} xt + h, xi+1, . . ., xn); we have
A{xtD)qithu - QithA(x,D)u = £ [gM ap (*)] Dprithu
\p\Z2m
and therefore
|| i4 (*, D) gM« - gMi4 (*, D) f*||flr-l(Rn ) ^ Cr || M ||fl2m + r-l(R,,).
+ +
Also
II ^/(y^) £*,/." -^,A(y>^)^llif2n,+r-Wy-l/2(Rn-l) ^ C,^ || M || H 2m + r- 1(R„ ) .
Finally, from (4.44), we deduce
(4.45) HeM«||fl2m + r-l(Rn)
^ Cr,e'{ll^,^l|if(Rn)xWn H2n, + r-n,y-l/2(Rn-l) + || ^ || H2m + r- 1 (R„ )} .
* y-o

146 4. A priori Estimates in the Half-Space
This inequality proves that Qih u remains in a bounded set of
H2m+r-1(Rn+) as h varies, with 0 < \h\ < (q0 - e)/2; but then we can
find a sequence h -» 0 such that Qih u -» # in H2m+r~1(Rn+) weakly;
du
but Qihu -> in @' (R"+), so that
dxt
— eH2™*-1^) for t = l,...,»-l
dxt
and furthermore its norm satisfies (4.45).
d ^/
To show that is also in H2m+r~1 (R+), we use the fact that
dxn
A ueHr (R+) and the ellipticity of A; indeed, we have a(0 0,2™>(0) 4=
=1= 0 and therefore also \a(0 o,2m){%)\ > e > 0 iov \x\ < q0 ii q0 is
small enough. Consequently,
d2mu 1 [ ^ )
—^ = —\A(x,D)u- X a9(x)B*u\eIF(R\)9
C*n #(0 0,2m)\x) \P\Z2m
Pn<2m '
3 u
from which we deduce that e H2m+r~1 (R+) and furthermore that
dxn
its norm in this space is less than or equal to
du II
f "-1
C\\\Au\\HriRn) + £
dxt
fl2m + r-l(Rn )
and therefore also less than or equal to the second member of (4.45).
Therefore i) is proved. □
2) To show ii), let us assume that
m-l
F = {/;g0. • • .^w-i}eI2(R"+) + ft #-2*+'"'+1/2(R''-1).
We consider the operator
^(0) = {Ao(0,D),BOtO(0,D) \t=o>...,Bm-uo(0,D) \tm0}
as a continuous linear mapping of
m-l
H2m{Rn+) into L2(Rn+) x n^2m"mi"1/2(Rn_1),
and its adjoint 0>*O) defined by
m-l
0>*oF = A*(O,D)?+ZBlo(OtD)(gj(y)®d(t)),
j = o
where A%{x, D) and B*t0(y, D) denote the formal adjoints of A0(x, D)
and Bjt0(y,D), j = 0, . . ., m - 1.

4.5 Half-Space for the Case of Variable Coefficients
We apply Theorem 4.1 and obtain
(4-46) ||F||L2(R-)xyH-^+v2(Rn-i) ^ \c M*(0, D) f +
147
m-1
+ T,Blo(0, D) gj(y) ® d(t)\\H-2m(Rn) +
j=o
m-1 |
+ ll/llfl-l(R») + Z \\Sj\\H-2m + mrl/2(Rn-l)\ ^
J = 0 J
But we see that the homogeneous part of degree m of A*(x,D) is
A*(x, D) and the homogeneous part of degree mi of B* (y, D) is B*t0 (y,D).
Thus we obtain that
II m-1
(4.47) L4*(0, D) / - A*(x, D) / + £ 5*o(0, D) (g,(y) <g) ,5(0) -
-E**(y.0)te;(y) «>«(<)) ^
J = 0 ||H-2m(Rn)
t^*(0,D)-^*(^,Z))]/ +
-1 I
+ E [B?.o(0,Z>) - Bj.0(y,D)](gj(y) <g> *(<))
j=o |
+ C,
H-2m(Rn)
<
H-2m + my-l(Rn)? ^
J = 0 J
(if q0 is sufficiently small, we may apply Lemma 2.3, thanks to the
hypotheses on the coefficients of A and Bj)
^ e |II/IIl2(R") + Z II &/
||jf-2m+m/(Rn)? "1"
{m-1
ll/IU-.(R.> + III&(y)®awil
H-2m+my-l(Rn)
(here we use the fact that, for s > 0,
\\gj(y) ® *WIIh-.(r-) ^ c kjllfl-.+i/2(R-i))
^ fi |||/IIl2(R+") + £ llgj||fl-2m + m/+l/2(R„-l)} +
!m-l J
II/IIh-1(R") + E ll^jllH-2m + my-l/2(Rn-l)|.
J = 0 J
Then (4.40') follows from (4.46) and (4.47) (choosing e = $). D

148
5. A priori Estimates in the Open Set Q
5. A priori Estimates in the Open Set Si and the Existence
of Solutions in //'(A)-Spaces, with Real s ^ 2m
5.1 A priori Estimates in the Open Set O
We return now to a boundary value problem {A, B}:
!A u = / in Q
BjU = gj on r, j = 0, 1, . . ., m - 1,
under the hypotheses of Section 1 on the open set Q and on the operators
A and BJt that is:
(i) Q is a bounded open set in Rn, with boundary r, an n — 1
dimensional infinitely differentiable variety, Q being locally on one side of J1;
(ii) the operator A is defined by
Au= £ (-I)'*" D>M(*),DV),
with apq e 2 (D) and is properly elliptic in D;
(iii) the operators Bj are defined by
Bju= £ bJh(x)Dhu
\h\gmj
with bJh e@(r), 0 ^ mj ^ 2m — 1, the system {B/JJpJo being normal
on r and covering A on r.
We consider the operator 0>t defined by
0>\ u-> 0>u = {Au\ B0u,..., Bm-i u}t
a continuous linear mapping of H2m(Q) into
m-l
L2(Q) x n H2m-mJ~1/2(r).
J=o
We denote by 0>* the adjoint of &> which maps
m-l
L2(Q)x n H~2m+mJ+1/2(r)
J=o
into (H2m(Q))' = HB2m{Rn) and is defined by
(5.2) <u,0*Fy = (&u,Fy(=<AuJy+mz<BJutgJ>)
m-l
VueH2m(Q)andVF = {f;g0t...,gm_1}eL2(Q)x n#~2m+m'+1/2(r),
j=o
where the brackets in (5.2) denote the appropriate dualities.

5.1 A priori Estimates in the Open Set Q 149
Theorem 5.1. Under hypotheses (i), (ii), (iii), for every integer r ^ 0,
we have:
1°) if
ueH2m(Q) and 0>ueHr(Q)x ]J H2m+r-mJ~l/2(r)t
j=o
then ueH2m+r{Q) and
(5.3) || W||H2»i+r(0) ^ Cr{|| 0* ^||H'"(r2)xmnH2m+r-my-l/2(r) + || U || H2m + r- l(fl)}
y-o
(Cr depending on r)\
2°) */
F = {/;g0. • • ..fo-i} e^2(^) xll jy-a-+^+i/2(JT),
J = 0
then there exists a C0, independent of F, such that
(5.4) || F || L2(fl)xWn1H- 2n,+»v+l/2(r) ^
j = 0
(m- 1 J
II^*^IIh^-(r-) + II/IIh-kr-) + Z kjllH-^^-iw) ,
ze>Aer£ / denotes the extension of f by zero outside of Q.
Proof. 1) Via "local maps" and "partition of unity", the usual
arguments bring us back to Theorems 3.1 and 4.3. Indeed, we may
take a finite covering of D by open sets {0J?L i, whose diameter will
be suitably fixed, and an associated partition of unity {0J?L i, dt e@> (Rn),
N
the support of 0t being in 0t and £0, = 1 in D.
i = l
N
Now, we show 1°). We have u = Yj®iu'> therefore it suffices to
t=i
consider 0t u for each i. Two possibilities exist: 0t r\ Q — 0% or (Pt n
r\Q cz 0it strictly.
In the first case, we apply Theorem 3.1; indeed if the diameter of 0t
is sufficiently small, there exists an infinitely differentiable diffeo-
morphism ipt of 0t into the ball a(q) of Rn; this diffeomorphism
transforms the spaces Hs{(9?), s ^ 0, into the spaces Hs{o{q)) (see Chapter 1,
Section 12.9), that is every element v e#s(0,) is transformed into an
element v* = v <> y)t e Hs((j(q)).
In the same way, A is transformed into an elliptic operator s/ of
order 2m in a(g).
Then, if q is sufficiently small, we may apply Theorem 3.1; it follows
that O'iu! (transform of 6tu) belongs to H2m+r(a(Q)) and
(5.5) ||0>'||h2—(„(e)) ^ Cr{||^(0;^)llH^)) + \\0iu'\\H^+r-Ha(6))}.

150 5. A priori Estimates in the Open Set Q
Coming back to the open set Oit we see that dtue H2m+r(Oi) and
therefore, thanks to the property of the support of 0t, 0tu e H2m+r(ii) and
I|0i«llfl2m+r(fl) ^ Cr{\\A{diU)\\HriQ) + || 0, « ||fl2m + r-l(fl)} .
But
A(Otu) =6tAu+ £ ypqD^uD^di with yMe9(D),
\p\< 2m
\q\ £2m
and therefore
(5.6) ||0£«||fl2«+r(O) ^ Cf{||M «||flr(0) + || 0f « ||n2m+r-1(Q)} .
In the second case (0£ n i3 c 0£, strictly), the argument is
analogous; we only need to note that we may define the diffeomorphism ipt
so that Oi n Q is transformed into the half-ball a+ (q) = {x \ x e a (q) ,
xn > 0} and @i n r into dxo+ (q) = {x \ x e a(g), xn = 0}, and that
we must also consider the operators 0#Jt the transforms of BJt j = 0,...,
m — 1. We may also assume that the coefficients of s/ and Ss are
extended to Rn and Rn_1 respectively, into infinitely differentiable
functions which, together with all their derivatives, are bounded. Then
(if the diameter of Ot is sufficiently small), we may apply Theorem 4.3.
It follows that 0i«eff2m+r(fi) and furthermore that
(5.7) IIMh^c^
{m-l \
\\QiAu\\Hr(Q) + X \\0iBJU\\H2n> + r-mrU2(r)+ \\0iU\\H2n>+r-HQ)\^
J = o J
m-l 1
A ^||flr(£) + 2] II^J W IIH2 «+'-'";-1/2 (D + IIWIIh2»i+i-1(0)}.
Summing over i, (5.6) and (5.7) yield
N
Il"llfl2». + r(fl) ^ X l|0*Wllfl2»»+r(fl) ^
i=l
^ Crdl^ttHflr^x^ilim+r-my-l^r) + II « ||fl2m+r-l(fl)}- □
7=0
2) Now we prove 2°) in analogous fashion. We use the same notation
as in 1) and the results of Chapter 1, Section 12.9 on the spaces Hs,
with s < 0.
If F = {f',g0,. . .,gm_i}, then
F-J^F, i.e. / = £0*/ and g, = £0^, / = 0,...,m-l.
J=l i=l i = l
Assume that (9tr\Q = Ot\ then ipt transforms 0tF = {0f /; 0,...,0}
into fl; F' = {0|/'; 0, . . ., 0}. If s/* denotes the formal adjoint of s/,

5.1 A priori Estimates in the Open Set Q 151
then s/* is the transform of A and we may apply Theorem 3.1; we
obtain
from which we deduce that
(5.8) ||H-2m(i?) + ||0j / ||h-1(^)}-
Now, assume that 0i n Q a 0i} strictly. Then y)t transforms 0t F
into OIF = {Oif;0igo,...,0'tgm-i}. We denote by 0*'{0\F'\ the
transform of 0* (0t F) and seek an explicit representation of 0**' (0J F') ;
we recall that 0>*(6tF) is defined (see (5.2)) by
(5.9) (u, 0* (fl, F)> = <4 «, fl, />+!<*, M, Si>
j=o
for all u e H2m {Q). Taking into account the property of the support
of 0| and therefore also of 0|, we see that 0>*' (0j F') is defined by
the brackets of the first member denoting the duality between H2m{R\)
and (H2m (R+))', and the brackets of the second member denoting the
duality between L2{R\) and itself and between H2m~mj~1/2(Rn-1) and
jj-2m+m.+1/2 ^n-ij ^ respectively. But, following the method of Section 4,
we may identify (H2m(R\))' with #j=n2m(R") and, denoting by uQ a fixed
extension of u' e H2m(Rn+) to H2m{Rn) and by v the extension by zero
for t < 0 of v e L2 (Rn+) to L2 (Rn), we have
<«', ^' (e; F')> = <^ ^o, e; /'> + !<#/ «i, e; g; ® a w>
J=0
= <«i, ^*(0;n> + Z <^o,«?(»;?; ® (5W)>,
the brackets, as usually, denoting the appropriate dualities. Therefore
P+'iP'tF9) is defined by
, > m-l
J = 0
and therefore coincides with the adjoint 0'* of the operator
0>'u = {^^;^0^lr = o>...,^m-i^1r = o}>
calculated on 0j F'.

152 5. A priori Estimates in the Open Set Q
Then we may apply Theorem 4.3, which shows that
II 9| F' llL2(Rn)x n H-2m+my+l/2(Rn-l) ^
J = 0
^ c ii^'MF)!!*--™ + ne;/'iu-.(R») +
m-1 j
+ 2j II ^< ^< HH-2W+»*i/-l/2(Rn- 1)1
But then, coming back to the open set 0t r\ Q t we see that
II 0* F\\L2(Q)xmjl H-2m+mj+l/2(r) =
7=0
< c
I^*(9i^IIh-2-(r-)+ II9i/IIh-kr-) +
m-1 |
+ Z l|S|firjllfl-2»»« + mri/2(r)}.
j=o )
and noting that
II^MIIn-»"«»> ^ C j||^*F|!fl-«(R-, + II/IIh-kr-) +
m-1 1
+ 2j II ^J IIH - 2m+ wy-1/2(^)1
it also follows that (5.4) holds. D
Remark 5.1. The estimates (5.4) are sometimes called the "dual
estimates" of the estimates (5.3) (see also the Comments at the end of
this chapter). D
5.2 Existence of Solutions in £F (&)-Spaces,
with Integer s ^ 2 m
We can now study the boundary value problem (5.1) from the
point of view of the existence of solutions in the spaces H2m+r(Q),
with integer r ^ 0. The essential aim is to show that the operator
(5.10) 0: u-> 0u = {Au\BQu9...tBm_xu}
is an indexed operator, mapping H2m+r(Q) into
m-1
Hr(Q)x [] H2m+r~mJ-1/2{r)
i = o
and then to express the compatibility conditions of the problem using an
adjoint boundary value problem.
We recall that 0 is an indexed operator if the dimension of the
kernel ker (0) of 0 is finite and if Im (0) is closed and its codimension

5.2 Existence of Solutions in Hs (Q) -Spaces
153
is also finite; then the index %{&) of 8P is given by
X {&) = dim ker (&)< - codim Im (0).
To show that 0, as defined by (5.10), is an indexed operator, we
shall use Theorem 5.1; from the "direct estimates" (5.3), we deduce
that ker(^) is finite-dimensional and that Im(^) is closed; from the
"dual estimates" (5.4) we shall deduce that the codimension of Im(^)
is finite.
We shall make use of the following lemma, due to Peetre [2]:
Lemma 5.1. Let E, F, G be three reflexive Banach spaces such that
E c F with compact injection and let <€ be a continuous linear operator
of E into G. Then the following conditions are equivalent:
1) the image under <£ in G is closed and the kernel of <€ is
finite-dimensional ;
II) there exists a constant C such that
(5.11) \\u\\E ^ C{\\Vu\\G + |M|F} V** e£<(1».
We are now in a position to prove
Theorem 5.2. Under the hypotheses (i), (ii), (iii), the operator SPy defined
by (5.10), mapping H2m+r(Q) into
m-l
Hr(Q)x YlH2m+r^~1/2(r)f r = 0,1,2,...,
admits an index %{&) which is independent of r\ the kernel of SP is equal
to the finite-dimensional space
(5.12) N = {u\ue^(D)t0u = 0}
(d)) Proof of the lemma: 1) Condition II) implies I). First of all, (5.11) implies
that the unit ball in the kernel E0 = ker (#) of # is compact; therefore E0 is finite-
dimensional. We decompose E into a direct sum E = E0 ® Ex.
The restriction of # to E1 is injective and we can show that
(0) \\u\\E^C\\Vu\\G ^uGE1
(by contradiction, using (5.11) and the hypothesis that the injection of E into F
is compact). It follows that Im(#) is closed in G, for, if t/j€Im(#) and tends
to v in G, then there exists Uj £ Ex such that # Uj = Vj and we deduce from (0)
that Uj tends to u € Ex and obviously # u = v, therefore v € Im(#).
2) Condition I) implies II). Again, decompose E into a direct sum E = E0 ®E1,
where
£0 = ker(#).
The restriction of # to E± is injective and surjective; therefore, thanks to the closed
graph theorem, we have (0). We can also show that
(00) \\u\\E<C\\u\\F Vw€£0
(still by contradiction, using the hypothesis that the injection of E into F is
compact). Then, if u € E and u = u0 -\- wx with ut€ Et, (5.11) follows from (0) and (00).

154 5. A priori Estimates in the Open Set Q
and the image under SP is given by the elements {/; g0,. . ., gm_i} of
m-l
H'{Q)x n H2m+r-m^-^2(r)
J=0
such that
fvdx+ £<g,,ft>=0
J = 0
f
I Q
(5.13) j ^or ^very element 0 = {v\ q>0,.. ., 9?m_i} o/ /A^ s/wctf
Lr = |0 6 I2 (fl) X [] i/-2m+mi+ 1/2 (jn) ^ ^* 0 = 0 ,
z^A^r^ ^* is the adjoint of SP considered for r = 0 {therefore SP* is given
by (5.2)).
Proof. For the time being, we denote by ^(r) the operator 9 considered
as a mapping of
m-l
H2m+r(Q) into i/r(i3) x [] ^2m+r"mj"1/2(^)-
J = 0
It follows from 1°) of Theorem 5.1 that the kernel of ^(r) is N, as given
in (5.12), for f] H2m+r(Q) = @(D) (see Chapter 1, Section 9.4), and
that r^°
!m-l l
Hr(Q)x [] H2m+r~ "«-1/2 (r) .
Applying Lemma 5.1 to ^(r) (the injection of H2m+r(Q) into H2m-r-l{Q)
being compact, see Chapter 1, Section 16), we deduce from (5.3) that N
is finite-dimensional and that Im(^(r)) is closed for all r.
We first consider Im (^(o>) i since Im (^(0)) is closed, the equation
m-l
&(0)u=F, with F = {f;g0,...,gm_l}eL2(Q)x f] H2"-*-U2(D
j = o
admits a solution if and only if
<F, 0} = 0,
« , > denoting the duality between
m-l
L2{Q) x [] H2m-mi~ 1>2 (r)
j = o
and its dual) for every solution of the equation 0>*o) 0 = 0, where ^*0)
is the adjoint of ^(0) and therefore coincides with the operator 0>*
defined by (5.2).
Therefore, we have shown that Im^(0) is the polar of the kernel
of &*.

5.3 Precise Statement of the Compatibility Conditions for Existence 155
We note that this follows only from 1°) of Theorem 5.1.
Now, we show that the kernel of &* is finite-dimensional and
coincides with the space JT. For this purpose, we use 2°) of Theorem 5.1.
In fact, thanks to (5.4), we can apply Lemma 5.1 to the operator ^*,
for the injection of
m-l m-l
L2{Q) x n H-2m+mi+lt2{r) into H~l {Q) x l\ H~2m+mi-^2(r)
j=0 j=0
is compact (see Chapter 1, Section 16).
Therefore, the kernel of &** (= ^*0)) is finite-dimensional.
Finally, we have to show that
codim Im(^(r)) = codim Im (^(0)).
But
m-l
Hr(Q) x n H2m+r-mJ-1I2{r)
codim Im(^(r)) = dim J~° //a . , r = 0, 1, . . .,
Im(^(r))
and we can easily see that the injection of
m-1 m-1
Hr(Q) x n H2m+r~mJ-l/2(r) into L2(Q) x ]J H2m~mJ-^2(r)
j=0 j=0
induces an isomorphism between the quotient spaces (with respect
to Im(^(r)) and Im(^(0)) respectively), for we already know that
Im(^(r)) is closed, that (5.14) is valid and that codim Im (^(0)) is
finite. Therefore codim Im (^(r)) = codim Im (^(0)) < +oo and the
theorem is proved. □
5.3 Precise Statement of the Compatibility Conditions
for Existence
We shall now state the compatibility conditions of the problem more
precisely by using Green's formula and the formal adjoint problems
introduced in Section 2.
Given A and the B/s, j = 0, . . ., m — 1, under the hypotheses (i),
(ii) and (iii), we may apply Theorem 2.1; therefore let us choose the
system of "boundary" operators {SjJ^Jb1 in such a way that Green's
formula (2.3) holds, that is (see also Remark 2.2):
/» /» m-1 r>
(5.15) (A u) v dx — u A* v dx = £ SjU CjV da —
q q r
m- 1 /•
,uTjvdo, ^u,veH2m{Q)i
m- l /»
j = 0 J
r

156 5. A priori Estimates in the Open Set Q
where the operators Cj and Tj} j = 0, . . ., m — 1, depend on A,
{Bj}fZo and {S^o, according to Theorem 2.1.
We first prove a "regularity" result which is analogous to
Theorem 5.1 i). We denote the scalar product on the space H2m~mr1/2 (jT)
by (. , )j and introduce the form
/» m-l
(5.16) \u,v] = AuAvdx + £ (BjU, Bj v)j
J j=o
Q
which is continuous on H2m (Q) x H2m (Q).
Applying (5.3) for r = 0 and Theorem 16.3, Chapter 1, we easily
obtain the existence of a constant c such that
(5.17) c-1 \\u\\2H2miQ) ^ [u,u] + ||«||l2(0) ^ c \\u\\l2miQ},
VveH2m(Q).
Let us then prove
Proposition 5.1. Let ueH2m(Q) and feHr(Q) be given (r being
a fixed positive integer) and satisfy
(5.18) [u,v] = \fvdx VveH2m(Q)',
h
then ueH+m+r(Q).
Proof. Choosing ve@(Q) in (5.18), we obtain A*Au = f; since
A* A is elliptic and of order 4m, we obtain from Theorem 3.1 that
u e H2m+r(o)) for any open set co such that co c Q\
We now use (as in the proof of Theorem 5.1 and in the same notation)
local maps and a partition of unity to reduce (5.18) to
(5.19) {u',0'v'} = j f'Fl/dx, W eH2m(a+(Q)),
where 0' is a C°°-function in o+ (q) which vanishes in a neighborhood
oi d2<y+ (q) = {x \ \x\ = q, xn ^ 0}, and where {u't v'} is defined by
/» m— 1
(5.20) {«', v'} = s/u' rfv'dx+ £ {@ju'>@jv'}j>
J j=o
{ , }j denoting the scalar product in
H2m~mJ-1/2(d1a+(Q)).
Thanks to (5.17), srf and Si satisfy the inequalities
(CT1 IKIll«(a + (W) ^ {"'•W'} + IWWUa^y =5
(5.21)

5.3 Precise Statement of the Compatibility Conditions for Existence 157
Using these inequalities, we can prove that 0' u' belongs to
HAm+r(o+ (q)) by the method of difference quotients. To simplify the
notation we shall write u,v,0, . . ., and set
(w, d) = uvdx
and furthermore use the notations qih u and rith u introduced in
Section 4.5.
For the time being, we admit
Lemma 5.2. Let she a positive integer. Then, for all q such that \q\ = s,
we have
(5.22) \{Q,ADqA6u)),v} - (-lY+1{u,Dl(dQu_hv)}\ ^
^ Ks |M|H2m(„ + ((?» I X HD>llH*"(a + (9))) »
Vu,veH2m(o+(Q)), D'yueH2m(a+(Q)), \p\^s,
where \q\ = s and Ks is independent of u, v and h.
From (5.19), for 1 $ i ^ » - 1 and h sufficiently small, we obtain
(5.23) {«,fle,.-»»} = (/,fle,.-»«') V»er((r+fe)),
hence
(5-24) l{«.0ei.-»»}lS*|f|
with K independent of v and h.
Using (5.22), with s = 0, we get
(5.25) | {Qith(0 u)tv}\^K*\\v\\H2mia+(e)) V* e H»»(a+ (q))9
where K* is independent of h and v. Hence, thanks to (5.21) and taking
v = qih (6 u), we obtain
(5.26) ^L e H2m(a+ (q)), i = 1,..., n - 1.
ox1
By iteration, we can show in an analogous manner that every
tangential derivative D%(0 u)t \q\ ^ r + 2m, belongs to H2m(o+ (q)).
Indeed, assume that this is true for 1 ^ \q\ < r + 2m. Then, using
(5.19) for i = 1, . . ., n — 1 and h 4= 0,h sufficiently small, and \p\ g r,
we get
{u,Dl(dQt,_hv)} = (f,D«y(6<>t,_hv)) = (-l)W(D'yf,DrP(6Qt,-hv)),
VveH2m(o+(e));

158 5. A priori Estimates in the Open Set Q
since / e Hr(a+(Q)) and \q\ < 2m + r, we have
consequently, using (5.22) for s = \q\, we obtain
where K* is independent of h and v. Therefore, using (5.21) again,
we have
(5.27) £—LeH**(a+(Q)), i=\,...,n-l,
dxt
hence
(5.28) D*(0 u) e H2m(a+ (q)) for \q\ ^ r + 2m.
Let us now use the fact that s/* s/(6 u) = / in o+ (q) ; writing s/ w
in the form
(5.29) sfw = £ <xp(x)D'u,
\p\ ^ 2m
we can always assume, thanks to the ellipticity of s/, that the coeffi-
d2m
(5.30)
cient of 2m in j/ equals 1. Therefore
***$*) = \m +
dAm (0 u)
\p\<2m y=l OXi\\q\^2m J)
where j8PtJta is of class C00 in or+(g). Then we have
(5.3D ^^ = /- I DPZ»
OXn \p\<2m
with
( feHr(o+(p)) and, thanks to (5.28),
(5.32) v+v*" V
I D«fo 6 L2(<r+ (e)) for |?| ^ 2m + r - 1.
It follows that
54m
(5-33) —^(D*y(6u)) = D«yf- £ D'Djg^ff-^+^cr+te)),
OXn \p\<2m
for |#| ^ 2 m + r — 1.
Therefore (5.18), (5.33) and Lemma 12.3, Chapter 1, applied to
Dqy{6u)} \q\ ^ 2m + r - 1, imply that
(5.34) ^(e^)ei/2m+1((r+(^)), for |?| g 2f» + r - 1.

5.3 Precise Statement of the Compatibility Conditions for Existence 159
But then (5.30) yields
DU,eH*(o+(Q)), \q\^m + r-2.
Then using (5.33) again, we obtain
(D*y(du))eH-2»>+2(o+(Q))} for \q\ ^ 2m + r - 2
dxn
and therefore, thanks to Lemma 12.3, Chapter 1, we have
Dqy(0u)eH2m+2(o+(Q))f \q\^2m + r- 2;
by iteration, we conclude that
due i/4m+r(or+ (g)), which proves the theorem. D
We still have to show Lemma 5.2.
We first show that if \q\ = s, then
(5.35) \(sf(euk (D}(0u))),s/v) - (-l)s+1(^u,s/(D«y(6Qit_hv)))\ ^
^K\\v\\H2mia+(J £ \\D>u\\H2mia+(eX
\\p\z* I
where K's is independent of u, v and h.
We recall that for every (p and ip in L2(o+(q)), we have
(5.36) (Qit* (0 p), y) = -(6 q>, Qtt_hip).
Using (5.29), we obtain
(5.37) s/(et.>(D}(6 «))) = e,.» J/(DJ(0 «)) -
- Z te(.»*,)0'T,.»(i>;(fl«))
|i?|S2m
= 5,^(0Z)J j/m) + eii»(j/(D;(e «)) -6D9ytfu)-
- S (e,.»«F)D'T,.lk(z);(fl«))
= ei.»(flD«j/ «) + •••
and therefore
(5.38) (^(e,.»(D;(fl«))),j/w) = (e,.»(e^j/«),^») + (...,j/i»),
and it is easy to see that we have
(5.39) \{...,stv)\^K's'\\v llH2n,((T + (e)) I 2* ll^y^llH2n,((y + (e))],
where 2?i' is independent of w, v and A.

160 5. A priori Estimates in the Open Set Q
Moreover, we have
(5.40) (st«,J*(D}(9Qt.-hv))) = (s/u,D'(0Qt,-ks/v)) +
+ (s/u, rf(D«y(0 9i,.h v)) - D«y(d Qi,.h sf v))
= {-\P{dDqys/u,Qlf_hs/v) + (s/u,...)
= (-i)l,|+,(e,.»(flZ);j/«).^») + (•*«,.••);
by integration by parts in the term (stf u,...) we get
(5.41) \(s/u,...)\£K:"M H2m(a + (Q))\ X \\DUWH2m(a + (Q)) .
Then (5.36) foUows from (5.38), ..., (5.41).
We use similar arguments for the terms
(5.42) {@j(Qt,h(D>)). @jv}j ~ (-1)|,|+1 {3j*. @j{Dty{eQl,-hv))}J,
taking into account the formulas
{QiA9V)>v}s = -{6(P>Qi.-hV>}j
and also using the trace theorem of Section 8, Chapter 1.
This completes the proof of the Lemma. D
From Proposition 5.1, we obtain
CoroUary 5.1. If in Proposition 5.1, we assume that fe@>(D), then
ue2{D).
Let us now recall that (see Theorem 5.2) the kernel of 9 is finite-
dimensional and is given by
N = {u | u e@(D), A u = 0, BjU = 0, / = 0, . . ., m - 1}.
We introduce the vector spaces
M = \v\veH2m(Q), j vudx = 0,V^eiVJ
H2Bm(Q) = {u\ueH2m(Q),BjU = 0, / = 0, . . ., m - 1},
H2cm(Q) = {u\ueH2m(Q),CjU = 0, / = 0,. . ., m - 1}.
We have
Proposition 5.2. // fe@(D), then there exists at least one solution
u e@(D) of the problem
\ A* u = f in Q
(5.43)
y CjU = 0 on r, j = 0, . . ., m — 1,
if and only if f el.

5.3 Precise Statement of the Compatibility Conditions for Existence 161
Proof, Green's formula (5.15) implies that the condition / e M is
necessary. Let us show that it is also sufficient.
First, we note that AT is a closed subspace of H2m(Q) and of L2{Q)\
M is also closed a subspace of H2m(Q) and moreover
(5.44) ||M||jf2m(G) ^ C[u,u], VweM, C independent of u.
We can indeed show (5.44) by contradiction. If we assume that
(5.44) is false, then there exist a constant C\ and a sequence uneM
such that
II «„ \\H2mm -> + oo and [un, un] £CX V».
Set
wn = .
II Un\\H2m(Q)
Then we have
(5-45) IKIIh^) = 1, V»,
(5.46) [wni w„] -> 0 if » -> + oo.
Thanks to Theorem 16.1 of Chapter 1 and to (5.15), there exists a
subsequence of wn, still denoted by wn, such that
\\u>m ~ wn\\L2(Q) ->0 if m,w-»+oo;
but using (5.17) and (5.46) it follows that
\\u>m - ^»Hfl2«(fl)->0 if W,W->+00
and therefore there exists a w e M such that
(5.47) || wn - w ||fl2m(fl) -> 0 if n -> + oo .
But (5.46) implies [z#, z#] = 0, hence z# e ker^ = N\ but we have
also shown that weM, hence w = 0, which is absurd since (5.45),
(5.47) imply that \\w\\H2m{Q) = 1.
From (5.44) and (5.17) it follows that in M, [u, u]1/2 is equivalent
to ||w||fl2m(0). But since v -> \ fvdx is continuous on M, there exists
a g 6 M such that g
[g,v'] = [fv'dx, Vv' etf.
We note that every v e #2m (i2) may be written
v = v' + v", v' e M, v" e N
(choose v" = projection of v on AT" in L2(Q)).

162 5. A priori Estimates in the Open Set Q
But [v"f v"] = 0 and f / v" dx = 0. Then
(5.48) [g,v] = [g,v'] = j fv'dx = j fvdx, VveH2m(Q)
and Corollary 5.1, shows that ge@(D).
Set u = A g and integrate by parts in (5.48) with v e2(Q)\ we
obtain A* u = f and
(5.49) j uA~vdx = j A*uvdx, VveH2Bm(Q).
But {B0 u, . . ., £m_i u, S0 u, . . ., S^! w} is a Dirichlet system of
order 2m on T. We can therefore apply Lemma 2.2 extended to Q and
show that as v describes HBm(12) n @{D), SjV describes 3)(r);
therefore (5.49) and (5.15) imply Cj u = 0, 0 ^ / ^ m - 1. D
Of course Propositions 5.1, 5.2 and Theorems 5.1, 5.2 remain valid if,
instead of problem {^4*, B}, we consider the adjoint problem {^4*, C}.
In particular, let us state the analogue to Proposition 5.2. We introduce
N* = {v\ ve@{D)}A*v = O.CjV =0,/ = 0,.. .tm - 1}
and
M* = \w\weH2m{Q)Awvdx = 0,VveiV*|.
We have
Proposition 5.3. // fe@(D), then there exists at least one solution
ue@(D) of
A u = / in Q
Bj u = 0 on T, j = 0, . . ., m — 1,
(5.50)
if and only if f e M*.
We can now show
Proposition 5.4. The vector space jV defined in (5.13) coincides
with the set described by {v, T0 v, . . ., Tm_1 v) as v describes N*.
Proof. Let us first show that if 0 = {v, q>0, . . ., ym_i} e^T, then
v e Sf (D). Indeed, if 0 e Jft then by definition we have
/» m-1
(5.51) Auvdx + Y<BjU,<pj} = 0, VueH2m(Q),
J j=o
hence in particular
(5.52) j Auvdx = 0, VueH2Bm(Q).

5.3 Precise Statement of the Compatibility Conditions for Existence 163
But (5.51), (5.52) (where us3(Q)) imply
(5.53) A*v = 0.
We decompose v = vt + v2i vl e N* = closed subspace of L2(Q);
we have
(5.54) \v2wdx = 0, VweN*.
Applying (5.15) to ueHlm(D) and to vx we obtain
f Auvxdx = 0, VueH2Bm{Q).
Then (5.52) implies
(5.55) f Auv2dx = 0, VueH2Bm(Q).
We shall now show that v2 = 0. Indeed, let us take A e 3)(D)\
we may write
A = A ux + *! with «! e #Bm(£) and At e N*,
since, if At = projection in L2(Q) of h on N*, we have
f (A - *i) ®dx = 0, V^eiV*.
Then Proposition 5.3 implies the existence of uleHBm{Q) such
that ^4 ^t = h — ht.
Furthermore, using (5.54), (5.55), we have
hv2dx = \ A u1v2dx + h1v2dx = 0.
Since A is arbitrary in ^ (£?), it follows that v2 = 0.
Consequently v = vt e2(D).
We can now apply Green's formula (5.15) to. j (A u) vdx in (5.51);
using (5.53), we obtain q
m- 1 m- 1 /•
(5.56) £<«,*'£/-7>>+ E 5^-C7^rfor = 0, VueH2m(Q).
j=o j=o J
r
Since the system {B0 u, . . ., £m_ t w, S0 u, . . ., Sm_ t u) is a Dirichlet
system of order 2 m on .T, we may use Lemma 2.2 extended to the
open set Q and therefore, if u describes 2(D), {Bj u, Sj ujfJo describes
(3{r))2m; then from (5.56) we deduce
TjV = <pjt CjV = 0, j = 0, 1, . . ., m — 1,
which proves the proposition. D

164 5. A priori Estimates in the Open Set Q
From Proposition 5.4 and Theorem 5.2, we finally obtain
Theorem 5.3. Under the hypotheses (i), (ii), (iii) and having chosen a
formal adjoint problem [A*, C} of {A, B} with respect to Green s
formula (5.15), the operator 0> = {^4; B0, . . ., Bm_l}, considered as an
operator of H2m+r(Q) into
m-l
Hr{Q)x Y\H2m+r-mJ-1/2{r), r = 0, 1,2,...,
j=o
has a finite-dimensional kernel given by the space
N = {u\ue@(D),A u = 0,Bou = 0, . . .,Bm_xu = 0}
and the image under SP is the subspace of elements {/; g0, . . ., g^.x} of
m- 1
Hr(Q) x Y[H2m+r-mJ-lf2{r)
which satisfy J=0
/» m— 1 /»
(5.57) \fvdx + % \gjTjvda = 0
o r
for every v belonging to the finite-dimensional space
N* = {v\ve®{D),A*v =0,Cov =0,...,Cm_1v = 0}.
Therefore, the operator SP is an indexed operator and its index %{&) is
given by
X(&>) = dimiV - dimN*.
Remark 5.2. The functions v and the operators Tj depend on the
choice of Green's formula, in such a way that (5.57) does not mean that
the image-space (which is obviously independent of Green's formula!)
depends on the choice of Green's formula. D
Remark 5.3. Denoting by H2m+r (Q)jN the quotient space of H2m+r{D)
by iV and by
\Hr{Q) x Y\H2m+r-mJ-^2{r);N*tA
the subspace of m_i
Hr(Q)x []^ir-m,-i/2(r)
defined by (5.57), and still denoting by & the operator after passage
to the quotient by N, Theorem 5.3 may be stated in the following form:
( the operator SP defines an [algebraic and topological) isomorphism
of H2m+r(Q)lN onto
m-l
(5.58)
{»!- 1 \
Hr{Q) x n#2w+r_Wj"1/2(^);^*><H
; = o J

5.4 Existence of Solutions in HS(Q)-Spaces, with Real s ^> 2 w 165
Remark 5.4. For elliptic boundary value problems, Theorem 5.3 yields
the theorem of the alternative of Riesz-Fredholm in the usual form
for elliptic equations of the second order (see Miranda [1]), for the
functions v of N* are the solutions in Q) (D) of the homogeneous formal
adjoint problem {^4*, C} with respect to Green's formula (5.15)
A* v = 0 in Qt CjV = 0, / = 0, . . ., m — 1.
5.4 Existence of Solutions in /75(£)-Spaces,
with Real s ^ 2 m
We shall give a first application of interpolation between Hilbert
spaces (Chapter 1) in order to study problem (5.1) in the spaces HS(Q)
with real s ^ 2w.
In fact, it is sufficient to interpolate between r and r — 1 in (5.58)
for fixed r ^ 0; also applying the results of Chapter 1, Sections 7 and 9,
we immediately obtain
Theorem 5.4. Under the hypotheses (i), (ii), (iii), the operator & defines
an [algebraic and topological) isomorphism of
!m-l l
Hs~2m(Q) x n#s"m'"1/2(^);N*><H
j = o J
for every real s ^ 2m.
Remark 5.5. The use of interpolation in Theorem 5.4 is not
indispensable; in fact, the same result can be obtained by applying the method
used in Sections 4 and 5 for the case integer s. A more interesting
application of interpolation theory will be given in Section 7. D
Remark 5.6. Even in the case s ^ 2w, we may still say that, in
HS(Q), 0> is an indexed operator and its index is still given by
%{&) = dim N - dim AT*
(invariance of the index by interpolation: Proposition 5.1 of Gey-
monat [2]). D
Remark 5.7. From the preceding theorems we deduce the fact that
if f e@(D) and gj g ^(jT), then the solutions u of problem (5.1) belong
to 3} (D). In Volume 3, we shall see that there is an even stronger
regularity: if r is an analytic (or Gevrey) variety, if the coefficients
of A and of the B/s are analytic (or of Gevrey class), / is analytic (or
of Gevrey class) in D, gj is analytic (or of Gevery class) on r, then u
is analytic (or of Gevrey class) in D. U

166
6. Application of Transposition
6. Application of Transposition:
Existence of Solutions in ZP(&)-Spaces, with Real s ^ 0
6.1 The Transposition Method; Generalities
In Section 5, we studied the boundary value problem (5.1) in
"regular" function spaces: the spaces Hs(Q)t with s ^ 2m. Now we
want to investigate what can be obtained from these results by
applying "transposition". We have already made use of the idea of
transposition in the search for "dual estimates" (see Section 4).
In this section we shall systematically develop this idea in order to
study problem (5.1) in "non-regular" spaces of solutions.
Many different exploitations of this idea of transposition are possible.
In the continuation of this chapter, we shall give a "natural" meaning
to problem (5.1) for u belonging to HS{Q)-spaces, with arbitrary real s,
with the g/s belonging to Hs~mJ~1/2(r) and f to a suitable, sufficiently
general space of distributions on Q. □
We still assume that the hypotheses (i), (ii), (iii) of Section 5 are satisfied.
According to Theorem 2.2, the system {CJJlTo covers A* and therefore
we may apply Theorem 5.3, taking A* instead of A and {Cj)™=q instead
of {Bj}?~J. In particular, denoting by Hlm+r(Q) the space
H2cm+r(Q) = {u\ueH2m+r(Q),CjV = 0,/= 0, . . ., ro - 1}, r ^0
and by {Hr(Q);N} the space
{H'(Q)',N} = lf\feHr(Q),hudx = 0 V«6M, r^O
' D J
and still denoting by A* the operator after passage to the quotient by N*,
we have
I ^e °fterat°r A* defines an (algebraic and topological) isomorphism
( ' j of H^m+r(Q)IN* onto {Hr(Q);N} for all real s ^ 0.
Therefore, we may consider transposing the isomorphism (6.1); but
for r > \y the dual of Hr(Q) is not H~r(Q) (it is not even a space of
distributions on Q). Thus, in order to obtain the desired result, we must
restrict the operator A* to a subspace of Hcm+r{ii); we choose the space
(6.2) Xr(Q) = {v\veH2m+r(Q),CjV = 0, / = 0, . . ., m - 1;
A*veHr0(Q)}, real r ^ 0.
With the norm of the graph
IMlko) = IMIfllm + rO,) + M WllitO),
it is easy to see that Xr(Q) is a Hilbert space.

6.2 Choice of the Form L 167
From definition (6.2) and (6.1), with A* still denoting the operator A*
after passage to the quotient by N* and [HrQ (Q) '> N} denoting the space
{Hr (Q); N} n Hr0 (Q), we deduce:
I ^e °Peraior A* defines an (algebraic and topological) isomorphism
( ] j of Xr(Q)IN* onto {Hr0(Q);N}, r^O.
This result is our starting point for the application of the method of
transposition. Indeed, by transposition, we deduce from (6.3):
Proposition 6.1. Assume that (i), (ii), (iii) of Section 5 are satisfied
and that r is real and ^ 0. Then, for every continuous antilinear form
vm -» L(vm) on Xr(Q)/N* there exists one and only one element u* in the
space {Hr0(Q))N}' such that
(6.4) <u-,1*7*} = L(vm) Vv# eXr(Q)IN*
and u* depends continuously on L (for the strong dual topologies), the brackets
in (6.4) denoting the duality between {Hr0(Q)\ N}' and {Hr0(Q); N}. □
It is immediate to interprete the space [HrQ (Q); iV}'; indeed, we have
(6.5) {Hr0(Q);N}'^H-'(Q)IN.
Therefore, also noting that (using (5.15))
w A* v dx = 0 VweN and veXr(Q),
we may say: given a continuous antilinear form L on Xr(Q)IN*, there
exists u e H~r(Q), determined up to addition of a function of N, such that
(6.6) {ufA*H} = L(vm) Vt># eXr(Q)lN*
and
Vv eXr(Q) belonging to v\
the brackets denoting the duality between H~r(Q) 'and Hr0(Q). D
Remark 6.1. Proposition 6.1, specified by (6.6), in a certain sense,
gives a solution of problem (5.1) in H~r(Q); but in (6.6) we have
"mixed" the equation A u = f and the boundary conditions Bj u = gJ}
j = 0, . . ., m — 1. Therefore, we now have to "separate" the two items
by choosing the form L in a suitable way and giving an interpretation of
the solved problem, once L has been fixed.
6.2 Choice of the Form L
Concerning the choice of L, it is natural to decompose it into two
forms, L = Lx + L2t in such a way that Lx gives rise to equation
A u = / in the sense of distributions on Q and L2 gives rise to the

168 6. Application of Transposition
boundary conditions BjU = gjt . . . in a sense to be specified in the
most natural possible way.
There is an "optimal" choice for L2, but a different situation is
encountered for the choice of Lx.
For the choice of L2, we must (recall Green's formula (5.14) and (6.6)!)
first consider the mapping
(6.7) v^STv = {T<>vt...tTm_xv)
as v describes Xr (ii). We can characterize the image J~{Xr (Q)) of Xr (Q)
under &~\ for, we have
Theorem 6.1. Under hypotheses (i), (ii) of Section 5 and if the system
{Bj}7=o is normal on T, v -+ ^ v is a continuous linear mapping of
m-l
Xr{Q) onto Y\ Hr+mJ+1/2(r) for all real r^O
J = 0
and there exists a continuous "right-inverse" of 3~.
Proof. 1) Applying the trace theorem for the spaces H2m+r(ii)
(Chapter 1, Theorem 8.3), we immediately see, Tj being of order
2m — ntj — 1, that v -» 3~ v is a continuous linear mapping of
m-l
X'(Q) into Y\Hr+mJ+1/2(r).
j=o
2) To show that &~ is surjective and the existence of a continuous
right-inverse of 3", we take an arbitrary \pj e Hr+mJ+1/2(r), / = 0,...,
m — 1. We have to solve the problem: construct v e H2m+r (ii) such that
(6.8) CjV = 0, TjV = <pj, j = 0,...,m—l,
(6.9) A*veHrQ(Q),
and such that {<pQ, . . ., <pm-i} -> v is a continuous linear mapping of
m-l
YlH'+»i+1'2(r) into H2m+r{ii).
j=o
If r ^ \> then condition (6.9) is a consequence of the fact that
veH2m+r{Q) and that HrQ(Q) = Hr(Q) (Chapter 1, Theorem 11.1). If
r > \y then (6.9) is equivalent (Chapter 1, Theorem 11.5) to
(6.io) yM*v) =0, * = o, i,...,|>-±;r((1)).
Since ^4* is elliptic, the system
{Cj.Tj.y,^*)}, / = 0,...,m-l, *' = 0,...,tf-i]-,
((1)) [a]"1 denotes the greatest integer less than <x.

6.2 Choice of the Form L
169
is a Dirichlet system of order 2m + [r — |]_1 + 1, with infinitely
differentiable coefficients on r\ thus, using Lemma 2.1 and the arguments
of Lemma 2.2 (in an obvious formulation, with Q instead of a+) it all
comes down to constructing weH2m+r(Q) with
yj w = y)j, / = 0, 1, . . ., 2 m + [r — £]~,
with y>j given in H2m+r~J~l/2 (r) and z# depending continuously on
the y)j's. This is possible thanks to Theorem 9.4 of Chapter 1. D
Choice of L2. Therefore, according to Theorem 6.1, we have an
"optimal" choice for L2\ more precisely
m-l
(6.11) L2(v) = I<g„ 7>>, with gi6 ff-r-.,-i/2(r)#
j = 0
where the brackets denote the duality between H~r~mrll2(r) and
Hr+mJ+1/2(r)',.L2> as defined by (6.11), is, for all r ^ 0, a continuous
antilinear form on Xr(Q), the g/s being arbitrarily chosen in
H-'-»j-u*(r). Q
Choice of Lx. For Llt we may in general proceed as follows: we
choose a Hilbert space (or, more generally, a topological vector space)
Kr{Q) of distributions on Q such that
f Xr(Q) aKr(Q) czL2(Q)
(6.12) . ...
[ with continuous injection;
. I @(Q) is dense in Kr(Q) (and therefore Kr(Q) is a normal
v ' j space of distributions on Q). □
Remark 6.2. Such spaces Kr(Q) always exist. For example, we may
take K'(Q) = L2(Q) or Kr{Q) = HS{Q)} 0 ^ s ^ | (for ^(fl) is
dense in HS(Q) if 0 ^ s ^ ^; see Chapter 1, Theorem 11.1). For further
comments on the choice of Kr(i2), we refer the reader to Section 6.3.
See also Problem 13.10 in Chapter 3. □
Since 2(Q) c Xr(Q), Xr(Q) is also dense in Kr{Q). The dual
K~r{Q) of Kr(Q) therefore may be identified to a subspace of 2' (Q)
and, if /eX"r(i3), the form
(6.14) L2(v)=«,v>
(where the brackets denote the duality between K~r(Q) and Kr{Q))
also defines a continuous antilinear form on Xr(Q).
Thus, consider the form
m-l
(6.15) L(v) = L,{v) + L2(v) = </,£> + Y,<gj, Tjv>.
j=o
If we make the convention that L(vm) = L(v) for every veXr(Q),
element of the class v* e Xr(Q)/N*, we see that (6.15) defines a con-

170
6. Application of Transposition
tinuous antilinear form v* -» L{v*) on Xr(Q)/N* if and only if
m-l
(6.16) </,*> + £<g;, 7>> = 0 Vv eN*.
j=o
Finally, we have obtained
Theorem 6.2. Under hypothesis (i), (ii), (iii) of Section 5, for all fixed
real r^O, if f e K~r(Q) and gj e H-r-mJ-1/2\r), /=0,...,m-l,
with (6.12), (6.13) and (6.16), there exists ueH~r(Q), determined up
to addition of a function of N, such that
m-i
(6.17) <«, 4* »> = </,*>>+ £<&,?>> V*eX'(12).
J = 0
Furthermore {/; g0,. . ., g:m_1} -* u* = u + N is a continuous linear
mapping of the subspace of
m-l
#-'(£) x n#~r~m'~l/2cn
wa^ «/> o/ elements satisfying (6.16), mfo H~r(Q)IN. U
Writing (6.17) for all ve@(Q), we obtain
(u,A*Hy =(f,vs> Vve@(Q)
and therefore u satisfies the equation
(6.18) Au = f
in the sense of distributions on Q.
We still have to specify the choice of / and to interpret the boundary
conditions "contained" in (6.17).
6.3 The Spaces 3s (Q) and D5^)
Problem statement
The problem solved by Theorem 6.2 is the more general as / is
taken in a "larger" space, therefore as Kr(Q) is "smaller".
We have seen (Remark 6.2) that spaces Kr(Q) exist, but we ignore
if there exists an optimum space Kr(Q) (i.e. the smallest possible). [The
difficulty is that if 2i (12) is dense in Fx (12) and F2 (Q) > it is not necessarily
dense in Ft (Q) n F2 (Q) provided with the sup-norm topology.]
Also note that the space Kr (Q) may depend on the boundary conditions;
thus, with the Dirichlet boundary conditions we may take Kr(Q)
= Hq{Q) (or, more generally, #ff+1/2(!2)) which is no longer possible
for other boundary value problems.
In what is to follow, we shall give a "universal" (i.e. independent of
the boundary conditions) choice for Kr(Q), while limiting ourselves
(in order to simplify the presentation) to the case where Kr(Q) is a

6.3 The Spaces 3S(Q) and DSA(Q) 171
Hilbert space (which is not mandatory!), which seems to be very "close"
to the "minimal" space valid for all boundary conditions (if such a
space exists!).
For this purpose we need the spaces Ss (Q).
.E5(f2)-Spaces, arbitrary real 5
We first introduce the definition for s = 0, 1, 2, . . .
Let q(x) e@ {D) and be positive in Q and vanishing on r of the
same order as the distance d(x, T) oi x to T
[i.e. lim eW x =^4=0);
there exist such functions, for r is an infinitely differentiable variety
(we have already introduced a function of this type in Chapter 1,
Section 11.2).
Definition 6.1. // s = 0, 1,. . ., we set
ES{Q) = {u\Q^D"ueL2(Q),\oc\ ^ s}$
provided with the norm
ll«lls.a» = ( I lle^^llLKO)) -
\|«|£s /
ES{Q) is a Hilbert space and we have
(6.19) S° (Q) = L2 (Q), H* (Q) c Ss (Q) c L2 (Q)
with continuous injection.
Proposition 6.2. Q{Q) is dense in 5s (Q).
Proof. Let 6V (x) be a sequence of functions of 3f (Q) such that dv (x) = 1
if d{x,r) ^ 2\v and dv(x) = 0 if d(x,T) ^ l/v and
1 v{ n ~ \d(x,r)\M
(Ca depending on be, but not on v); such a sequence exists. Now, let
u eSs(Q)'t then dvu e HS(Q) and has compact support in Q. We verify
that
dvu -»u in SS(Q) as r->+oo.
For this purpose we have to show that (f"lD"(dvu) -» Q^^Dau in
L2{Q)\ but
dvQWD"u->QWD"u
in L2(Q); therefore, it is sufficient to show that
^(Dpd9) (Dyu) -»0

172 6. Application of Transposition
in L2(Q), \p\ ^ 1, \y\ + |0| = |a|, v-> + oo. But gW(Z>M,) (Dy«)
vanishes for d(xtT) ^ 2\v (for |/?| ^ 1) so that, according to Lebesgue's
theorem, we have the desired result if we note that
\Q^{D^dv){Dyu)\ = \Q^(D^dv)\\Q^Dyu\ ^Cp\Q^Dyu\eL2(Q).
Therefore now it suffices to approach dvu (fixed v) in Hs(ii) with
functions oiSf{Q); this is possible via the usual regularization procedure,
for the support of dvu (v being fixed) belongs to a compact set in Q. D
Definition 6.2. Let real s > 0 not be an integer, s = k + 6, wztfA
integer k ^ 0 #w^ 0 < 9 < 1; w setf
From this definition, (6.19) and the inclusion properties of Hs(ii),
there results that
(6.20) Hs (Q) c £* (fl) c 3s' (Q) c L2(Q), s, s' real and > 0, s' < s.
From Proposition 6.2 and the density theorem (Chapter 1,
Section 2), we also deduce that
(6.21) 2(Q) is dense in SS{Q) for all real s ^ 0.
Therefore, we see that Ss (Q) is a normal space of distributions on Q;
its dual may be identified to a space of distributions on Q. We denote
this dual space by E~s(i2), i.e. we set
(6.22) S-S{Q) =(SS(Q))', s> 0.
Proposition 6.4. For integer s > 0, every feS~s(Q) may be
represented (non-uniquely) in the form
(6.23) / = £ D« (eW /J , mtt /a e L2 (fl).
|«|£s
Proof. According to the Hahn-Banach theorem, every continuous
linear form on SS(Q) may be written
M(<p)= I \g^a{D'vdx gaeL2(Q).
According to Proposition 6.2, M is defined by its values for each
(pe@{Q), from which (6.23) follows if we set fa = (-1)1*1 ga. D
Choice of the space Kr(ii)
For all real r ^ 0, we take
Kr(Q) = S2m+r(ii).
This choice is permissible, for (6.12) is verified according to (6.20)
and (6.13) according to Proposition 6.2. □

6.4 Density Theorem 173
Then Theorem 6.2 may be stated as follows: denote by DJr(Q) the
space
(6.24) D2r{Q) = {u \ u e H~r(Q), A u e S-2m-r(Q)}, r ^ 0,
with the norm of the graph
IMlD-r(fl) = ||«||fl-ra,) + \\Au\\l-2m-rW.
A
Then, we have
Theorem 6.3. Under hypotheses (i), (ii), (iii) of Section 5, for every fixed
real r ^ 0, if feS-2m-r(Q) and gjeH~r-mJ-1/2 (r), j = 0,... ,tn - 1, with
(6.16), there exists u e D2r '(D), determined up to the addition of a
function of N, such that
m-l
(6.25) <«, A^} = </, v> + £ <gj, 2>> V* 6 X>(fl).
j = o
Furthermore, {/; g0, . . ., gm_x} -* u* = u + N is a continuous linear
mapping of the subspace of
m-l
(6.26) 5-2"-r(£) x fl ^-r-m^1/2(r)
m#^£ of the elements satisfying (6.16), into H~r(Q)/N.
To go further with the interpretation of the result we must now give
trace theorems for the elements of D2r(D)', we shall do this in the
following sections.
6.4 Density Theorem
Theorem 6.4. Under hypotheses (i), (ii) of Section 5, the space 2(D)
is dense in Djr(D) for all real r ^ 0 with r — \ not an integer.
Proof. Let u-*M(u) be a continuous linear form on D2r(D): it
may be written
(6.27) M(u) = </, u} + <g, A «>, with / e ff£(fl) and g eE2m+r(D),
since the intervening spaces are Hilbert spaces and therefore reflexive.
Suppose that we have
(6.28) M((p) = 0 for all ye 2(D).
We have to show that under these conditions
(6.29) M{u) = 0 for all ueD~Ar(D).
But every 99 e 2(D) is a restriction to £? of a function 0 e 2 (Rn).
We denote by / and g the extension to R" of / and g by zero outside Q
and by si an operator extending ^4 to R" in the following sense: si is
a linear operator of order 2m with infinitely differentiable coefficients

174
6. Application of Transposition
in Rn, which coincides with the operator A in D and which is properly
elliptic in 0, where (9 is a bounded open set with infinitely differentiable
boundary d®, with D a (P. Of course, there exists such an extension
of A. Also note that / and g belong to L2 (Rn).
Therefore (6.28) may be written
(6.30) M(q>) = </, <Z>> + <f, sf 0} = 0
where the brackets are taken in the sense of distributions on Rn; so
if si* denotes the formal adjoint of si, we have
(6.31) rf*g = -/
in the sense of distributions on Rn.
But, since r — \ is not an integer and f e Hr0 (Q), according to
Theorem 11.3 of Chapter 1, we have f eHr(Rn) and therefore its
restriction to 0 belongs to Hr(0). We would like to show that g \e e
eH2m+r((P) foUows.
Let us show that there exists weH2m+r(&) nH$(&) such that
(6.32) si* w = -/.
Using Theorem 5.4, applied to the open set 6 and to the Dirichlet
problem si* w = —/, yjW = 0, / = 0,...,m— 1, such a w exists
if and only if
(6.33) j fzdx = 0
o
for all ze@)((9) such that si z = 0 in 0 and yj z = 0, / = 0,... ,tn - 1
on d(9.
We shall prove that this condition is satisfied: let 6 be a function
of @{<9) equal to 1 on D; then
f fzdx = f JWz dx = - \ (si*g)lTzdx = - \g si{6z) dx = 0,
0 0 0 0
for g = 0 outside i2 and 6 = 1 on D.
Therefore w exists and
w - | e L2 (0) with j/*(w - g) = 0 in (P.
According to the hypoellipticity of si* (Theorem 3.2), it follows that
w — g is infinitely differentiable in 6 and therefore, since w e H2m+r((P)
and g vanishes outside Q, we see that g e H2m+r (0), whence (since g
vanishes outside Qj) yjg = 0, / = 0, . . ., [2m + r — J]", on r and
therefore (Chapter 1, Section 11) geHlm+r(Q).
Now, we compute (g, A u) with ^ e D2r(Q)', since ^(i3) is dense in
Hlm+r(Q), there exists a sequence ^fc e 9(G) with ^fc -> g in Hlm+r(Q);

6.5 Trace Theorem and Green's Formula for the Space DSA(Q), s 5^0 175
so that we have
<g, A u} = lim <gk, A u) = lim {A* gk, u}
k-*co k-*ao
= (A*g,u}t
for u e H~r(Q) and A* gk -» A g in Hq(Q). Therefore
M(u) = {A* g + /, ^> = 0, for 4* g + / = 0 according to (6.31). D
Remark 6.3. In the proof of Theorem 6.4, an exception appears for
the parameter r: r — \ — integer; but we shall see further on
(Remark 6.5) that the theorem holds also when r — \ is an integer. D
6.5 Trace Theorem and Green's Formula
for the Space DSA(Q)9 s ^ 0
Theorem 6.5. Under hypotheses (i), (ii) of Section 5 and if the system
{Bj}?=o is normal on r, for all real r ^ 0, with r — \ not an integer,
the mapping u -> B u = {B0u, . . ., Bm_l u] of 2{D) into {@{r)}m
extends by continuity to a continuous linear mapping, still denoted u-*Bu,
of
m-1
D7(£) into YiH~r~mj~1I2(r)-
j=o
Furthermore, for u e D"2r{Q) and v e Xr(Q), we have "Green's formula":
m-l
(6.34) (A u, v} - <«, A*~i} = - £ <B, u, 7>>,
j=o
where the first brackets denote the duality between S~2m~r (12) and 52m+r (Q),
the second between H~r(Q) and Hr0(Q) and (BjU,TjV) the duality
between
H-r-m'-lf2(r) and Hr+mJ+1^2(D.
Proof. Let u be given in D2r(0) and let <pj, j = 0, . . ., m — 1,
be given in Hr+mj+1/2(r). Consider the right-inverse of F introduced
in Theorem 6.1 and apply it to {q)0, . . ., <pm-i}\ we obtain a function
V(pEH2m+r(Q) such that
Cj v9 = 0, Tj v(p = yj, j = 0,. .., m - 1 and A* v^ e Hr0 (12),
Vy depending continuously on the (p/s. Then we may consider the
expression
(6.35) Z(v9) = (u, I%> -(Au.vJ,
where the first brackets denote the duality between H~r(Q) and Hr0(Q)
and the second between 3"2m~r (Q) and S2m+r (Q) (note that v9 eXr(Q) c
czS2m+r(Q)).

176 6. Application of Transposition
Z(v9) is independent of the "right-inverse" used; indeed, if vx
and v2 are two functions such as v9, then x = vi ~ vi satisfies the
conditions
yj% = 0, j = 0, . . ., 2m — 1, if r ^ \,
j = 0, . . ., 2m + [r — -J]" if r > \
(we use the same considerations as in the proof of Theorem 6.1, 2)).
Therefore (Chapter 1, Theorem 11.5)
x 6 Hlm+r (Q), so that (u, Z^> = {A u, x>
and therefore Z^) = Z(v2).
Therefore (6.35) depends only on q> and we may write Z{q>) instead
of Z(vv). Using (6.35) for Z(q>) we see that 99 -> Z(y) is a continuous
antilinear form on
m-l
therefore
m-l
(6.36) Z(<p) = £<t, «,£,>,
where r, m e H"r"mJ'112 (f).
Still using (6.35), we easily see that the mappings u-+TjU are
continuous linear mappings of DJr(Q) into H~r~mJ~1/2(r).
Now, we shall verify that
(6.37) XjU = BjU for « e ^(D).
If we take <pj e ^{F), j = 0, . . ., m — 1, we may construct the right-
inverse v in Q}{U) (see the proof of Theorem 6.1 and Lemma 2.2);
then, according to Green's formula (5.15):
m- 1 /• m- 1 /»
Zfa) = £ £^7><*<r = £ (Bju)<pjdo V^e^(r),
j = oJ j=o J
r r
whence (6.37), comparing with (6.36).
Also note that the mapping tj is uniquely determined by the operator
Bj, for, thanks to Theorem 6.4, 2f\Ji) is dense in D'2r{Q).
Finally, Green's formula (6.34) results from the preceding
considerations: for if v eXr(Q), then q>j = TjV e Hr+mJ+1/2(r) and we may
take Vy = v in (6.35) and therefore (6.34) follows from (6.35) and
(6.36). D
Remark 6.4. If r — \ is an integer, the proof may still be carried
out: we obtain the existence of a continuous linear mapping u -> x u

6.6 Existence of Solutions in DSA (Q)-Spaces, with Real s <J 0 177
= (r0 u, . . ., rm_! u) of D][r(Q) into
m- 1
HH-r-mJ-i/2(r),
J = 0
which coincides with B u =-- {B0 u, . . ., £m_ t u] on & (D). But we can
not say that r u is the extension by continuity of B u and therefore
that it is uniquely determined by B u, for we do not know whether
2(D) is dense or not in D2r(0)- Nevertheless, we shall see further
on (Remark 6.5) that Theorem 6.5 also holds for r — \ = integer.
In this case we may still write Green's formula (6.34), but with the
operator xs instead of B5.
6.6 Existence of Solutions in DSA (£)-Spaces, with Real s ^ 0
We are now in a position to interpret Theorem 6.3 in a more
precise way.
Indeed, we may write either formula (6.25) or (6.34) for the solution u
obtained in Theorem 6.3. But we already know (see (6.18)) that A u = /;
therefore, it follows that
m- 1
Z<BjU-gj,Tjvy = 0 VveX*{Q)
and, thanks to Theorem 6.1:
m- 1
I <Bj u~gj, <Pj> =0 V^ 6 #'+»'+ 'l2 (H
and therefore B}u = gJ} j = 0, . . ., m — 1.
Therefore, we have shown
Theorem 6.6. Under hypotheses (i), (ii), (iii) of Section 5, for all real
s ^ 0, with s — \ not an integer\ the operator
0>\ u -» S9u = {A u9B0u, . . ., Bm_l «},
defines an (algebraic and topological) isomorphism of DsA(Q)jN onto the
space
{m-l
ss~2m(D)x n Hs-mJ-^2(r)]N*>tr
of elements {/; g0, . . .,Zm-i} belonging to
m-l
Es-2m(Q)x [] Hs-mJ-l'2(D
J=o
and satisfying (6.16).

178
6. Application of Transposition
Remark 6.5. Let us look at the case s — \ = k, integer k ^ — 1.
Using Remark 6.4 and the same arguments as for Theorem 6.6, we obtain:
under the hypotheses (i), (ii), (iii) of Section 5, the operator
u -> 0>xu = {A u;r0u, . . .,rm_! u)
defines an [algebraic and topological) isomorphism of DkA+1/2 (Q)/N
onto the space
m-l
(6.38)
!m-1 1
Sk+1,2'2m(Q)x n Hk-mJ(r);N*,r\
j = o J
o/ elements {/; g0, . . ., g^} belonging to
m-l
J = 0
and satisfying (6.16).
On the other hand, using Theorem 6.6, we obtain that ^ -» SP u is
also an isomorphism of
(6.39) DkA(Q)/N onto |s*-2,»(fl) x [] Hk'mJ-^2(r); N*,3r\
and of
(6.40) DkA+1(Q)/N onto pfc+1-2m(i3) x [] ^fc+1-m^1/2(r); iV*, ^[.
By interpolation, there results that 3P is an isomorphism of
(6.41) [DkA(Q)IN,DkA+1(Q)IN]l/2 onto
m-l 1
,s*-2,"(fl)x Y[Hk-mJ-if2(r)\N*9^\9
J=0 J
m-1 '
sk+1-2m(Q) x n#*+1~m'~1/2(^);N*,jr
J = 0
1/2
It is easy to verify, by one of the (equivalent) definitions of the spaces
[X, Y]e given in Chapter 1, that
(6.42) [Xt xX2>Y1x Y2]e = [XltX2]9 x [Ylf Y2]9, 0 < 6 < 1,
if {Xi} YJ is a couple of Hilbert spaces having analogous properties to
the couple {X, Y}. Of course, this formula is valid for a product of
any finite number of spaces. Therefore, using the results of Chapter 1,
Section 13.4, the definition of SS(Q)-spaces, the duality theorem
(Chapter 1, Section 6.2) and the interpolation between the spaces
Hs(r), we deduce from (6.41) that 9 is an isomorphism of
(m-l \
Sk+1/2~2m(Q)x Y\Hk-mJ(r);N*>tT .
j = o I

6.6 Existence of Solutions in DSA(Q)-Spaces, with Real s<>0 179
Therefore, we have the following situation:
& and ^T are two isomorphisms of
([DkA(Q),DkA+l(Q)]1/2)IN and DkA+l>2(Q)IN
respectively, onto the same space ls*+ 1/2~2m (Q) x J] Hk~mJ (r); N*, y\.
But it can be shown, directly and easily, that
(6.43) [DkA(Q),DkA+1(Q)]1/2 c Di+,/2(0).
Furthermore, 0(£) is dense in [0$(fl), D$+1(fl)]1/2, thanks to the
density Theorem 6.4 and to Chapter 1, Section 2.1, and on £${D)
& u = 0>ru (see Remark 6.4).
Therefore
(6.44) [D* (Q), D\+»(Q)] 1/2 = D\+"2 (Q)
and 0> u = &T u on this space.
Therefore, even if s — \ is an integer, the density, trace and existence
theorems (Theorems 6.4, 6.5 and 6.6) are valid.
Finally, we may state
Theorem 6.7. Under hypotheses (i), (ii), (iii) of Section 5, for all real
s ^ 0, the operator
0>\ u -+ 0>u = {Au,B0u, . . .,£„,_! u)
defines an [algebraic and topological) isomorphism of DsA{Q)jN onto the
space \ss-2m{Q) x n Hs-mj-1/2(r); N*, s\ of elements {/; g0>---,gm-i}
m-l
belonging to Ss-2mx ]J Hs-mrll2{r) and satisfying (6.16).
j = o
This is the extension of Theorem 5.4 to HS(Q)-spaces, with s ^ 0.
We may again say that, in DSA (Q), & is an indexed operator and
that its index is given by the formula %{&) = dim AT" — dim AT"*.
Note that we have solved problem (5.1), that is
(6.45) Au = f in Q,
(6.46) BjU = gj on T, j = 0, . . ., m — 1,
the equation (6.45) being satisfied in the sense of distributions on Q and
the boundary conditions (6.46) in the sense of the trace Theorem 6.5;
therefore, we have given a "natural" meaning to problem (5.1) in the
spaces HS(Q) with real s ^ 0 (see also Section 8.1, below, for the
interpretation of (6.46)). □

180
7. Application of Interpolation
Remark 6.6. Theorem 6.7 solves the boundary value problem (6.45),
(6.46) with arbitrary distributions gj on /'(and /, for example, belonging
to a space SS(Q), arbitrary and fixed).
Indeed, r being compact, we have
\jH-{r) = S'(T).
Therefore, if we want to solve problem A u = /, Bj u = gj} j = 0,...,
m — 1, with / given in a space 5s (Q), arbitrary and fixed, and gj given
in 2' (r), there exists /u > 0, sufficiently large, such that / e S~ti~2m(Q)
and gjeH~ll~mrll2{r) and therefore we may apply Theorem 6.6 and
find ueH-**{{}).
Note that (J #""(£) c 21 (Q), strictly. But we shall see in Volume 3
s>0
how this may be further extended.
7. Application of Interpolation: Existence of Solutions
in jfiP(&)-Spaces, with Real s9 0< s < 2m
7.1 New Properties of .Es (&)-Spaces
We shall first introduce some new properties of the spaces SS(Q)
of Section 6.2.
Proposition 7.1. For integer s > 0, 5s (Q) coincides with the space
of u's belonging to 2'[Q) such that
(7.1) QsueH50(Q).
Proof. 1) Let ueSs{Q)\ then we see that QsueHs(Q). Indeed,
using the formula of Leibniz, we have
(7.2) D«(qsu)= X L7(x) q-MD"-v u
where LY (x) is a continuous function on D; therefore for | oc | ^ s, we
have
D" (qs u) e L2 (Q), i.e. QsueHs (Q).
Furthermore, qs u may be approximated with functions of Si (Q) in
HS(Q)', indeed, there exists (Proposition 6.1) a sequence <pv e S(Q) such
that <pv -> u in ES(Q), i.e.
in L2(Q), for \oc\ ^ s; again using (7.2), it follows that
Qs <pv -* Qs u
in #s (Q); but Qs(pveS (Q), therefore qs u e Hs0 (Q).

7.1 New Properties of 3s (Q) -Spaces 181
2) Conversely, let u e 9)' (Q) with (7.1), we show that
(7.3) Q^D"ueL2(Q) V* with \<x\£s.
Via "local maps" and "partition of unity" we are led to the
following situation: we have a function v(y,t), which is locally in L2(R+)
and has compact support in R"+, such that, for fixed r, with r = 0,l,...,s,
and fixed Dvy, with \y\ ^ r, the function w(y,t) = t5 D^v(yf t) belongs
to L2(Rn+) as well as its derivatives D{w for / = 1, . . ., r — \y\ and
furthermore Djw(y, t) |r=0 = 0, / = 0, . . ., r — \y \ — 1. Then, we
may apply Lemma 10.1 of Chapter 1 to obtain
+ 00
(7.4) j t-^-^WwiyJJWl^-^dtS
0
+ 00
0
+ 00
= C j t-^-h\-u \\Dtw(y,t)fLHRn.l)dt ^
0
+ 00
S C2 j #-2<r-|y[-i> \\tDtw(y,t)\\2LHRn.1)dt ^
0
+ 00
^ . ^ C-* j r2 \\Drt-M-1 w(y.t)\\UR-i>d* ^
0
+ 00
£C> \t-*\\tDrt-\v\w{yJ)\\2L2{J,n-^dt
= ll^r-,y|^l|L(Rn)< +00.
Therefore
(7.5) l^-r+JDJt(tsDvyv)eL2(Rn+)
for r = 0, \, . . ., s, \y\ ^ r, / = 0, 1, . . ., f — \y\, from which we
obtain
^^^D^6l2(R"+) for j + \y\^s,
which proves (7.3). D
We are now in a position to prove
Theorem 7.1. // integer s > 0, we have
(7.6) [S* (Q), L2 (Q)]e = {u\ue&(Q), q*1-^ u e [Hs0 (Q), L2 (Q)]e}.
Proof. — 1) Let u e [5s(Q), L2(Q)]e'} then according to the holo-
morphic interpolation (Chapter 1, Section 14), there exists / such that

182 7. Application of Interpolation
u = f(0), where f -> / (f) is a continuous function taking its values in
L2(Q) in the strip
* = {CIC = f+ iq,0^f ^ 1},
of polynomial growth in r\ (Chapter 1, Corollary 14.1), and holomorphic
taking its values in L2{Q) in the interior of J*, rj -> f(irj) being
continuous, of polynomial growth in rj, taking its values in 3s (Q), rj ->
-> / (f + i ?y) being continuous, of polynomial growth in rj, taking its
values in L2 (Q).
Consider the function
(7.7) C^dC)=es(1-°/(C), Ze@
and let us verify that
f f -> g(f) is a continuous function, of polynomial growth, of ^
I into Z,2(i2) and analytic in the interior of J*.
j *7 -> g (i ?y) is a continuous function, of polynomial growth, of R
( 9) [ into HS0{Q).
First of all g is continuous, for
k(C)llL2(c)^c ||/(o ||L2(C).
It is also analytic in the interior of ^; in fact, it suffices to apply a
theorem of Grothendieck on holomorphic functions taking their values
in a topological vector space (see Grothendieck [1]) and to verify that
C^<g(C),?> = jg(C)<pdx
for all <p G Of (D), is a scalar holomorphic function in the interior of £8.
But
<g(Q><P>=<f(Z),Qs(1-°<P> and C^f(C) and £->?«-* <p
are holomorphic with values in L2(Q); therefore, C-»<g(C),<p> is
holomorphic and (7.8) is proved.
To verify (7.9), using Theorem 11.8 of Chapter 1, it suffices to verify
first that
(7.10) Q-s+]«\D«g(irj)eL2(Q) V^eR, |*| ^ s.
But, by the formula of Leibniz, we have
e-+l«lD*g(iq) = e-+W £ Lfi{x9y) g*1-1^-^^-'/^)
P
{O^PiZoa}
where the Lp(x,rj) are polynomials in rj, with coefficients which are
continuous functions of x in U. But / (i ?y) e £* (Q) and therefore

7.1 New Properties of E*(Q)-Spaces 183
^-\P\Da-pf(itj) eL2(Q) and therefore (7.10) is proved. Furthermore,
we obtain
||e-'+l»l0-g(ii7)||1HO) ^ P(V) ||/(ii?)||«i»
with P(?y) a polynomial in rj of degree |<%|. Therefore (7.9) is also proved.
It follows that
that is
Q5(1~e) f(0) = Qs(1-6) u e [Hs0 (12), L2 (Q)]9.
2) Conversely, let u e 9' (Q) with qs^-0) u e [Hg0 (Q), L2 (Q)]d; then,
still using the method of holomorphic interpolation, we see that q5(1~9) u
= y>(0)> where f -> y(f) is a function taking its values in L2(Q),
continuous in J*, of polynomial growth in rj and holomorphic in the interior
of J*, rj -> y> (i ?y) being continuous and of polynomial growth in rj,
taking its values in HS0(Q).
Consider the function
(7.11) Z^x(0=Qs«-i)V>(Z), te@
and verify that
C -* X (C) is a continuous function of J* into L2 (12) of poly-
1 nomial growth in ?/;
(7-13) , .._ T2
f ~* X (C) is holomorphic in the interior of ^ (with values
in L2 (12));
f ?7 -> % (i ?/) is a continuous function of R into iJ5 (12) and of
1 polynomial growth in rj.
We verify that (7.12) holds. We note that not only ip(0) belongs to
[HS0(Q),L2(Q)]9, but also
W(6 + irjo)E[Hs0(Q)tL2(Q)]et Vq0.
Indeed (and this is a general property on [X, Y]0-spaces) it suffices to
not that the function ip (f) = ip (f -f i ?y0) satisfies exactly the same
conditions as ip(£) (i-e- is a continuous function of & into L2(Q), of
polynomial growth in ?y, holomorphic in the interior of &, rj -> y> (i ?/)
being a continuous function of R into #o (-0) of polynomial growth in ?y);
therefore yi(0) = ip(6 + itj0) e [^S (12), L2 (12)]0. Using Remark 11.9 of
Chapter 1, for ip (f) with f = 6 + i ?y, we obtain
e"s(1"Re%(C)eI2M VCe^
and therefore that %(£) eL2(Q) and
IIz(C)IIlho) = lle1<R,:"l)v(f)llL»o» < +°°. vce^,
whence (7.12), thanks to the properties of ip.

184 7. Application of Interpolation
To verify (7.13) we use the theorem of Grothendieck [1] again;
the scalar function C-» <# (C), 9?>, V(pe@(Q), is holomorpbic in the
interior of S, for (x(0> <P> = <W(0>Qs(C~1) <P> and f -> y(f) and f ->
-» £S(C-1) 99 are holomorphic and take their values in L2{Q). Finally,
we verify (7.14); for \<x\ ^ s, we have
Ql^D«x(iri) = e'aIZ)a(e,<i,»-1>y(iq))
= eH I L'fi(x,r,)Qi'''-'-MD*-eV(ir,),
P
where the Lp(x, rj)'s are polynomials with respect to rj with coefficients
which are continuous functions of % in D. But y>(i ?/) e #0 (-0) an(i
therefore (Theorem 11.8, Chapter 1) Q-s+\"\-W\D"-p ip(irj) eL2(Q), therefore
g^D«x(iri)eL2(Q)} i.e. x(ir,)e&(Q).
Furthermore
We^D'xiirih^^PWWfiiriWwn for 1*1 ^s,
where P(?y) is a polynomial in ?y and therefore (7.14) is verified.
From (7.12), (7.13) and (7.14) we deduce that
X(0) = ef-VviO) = e'<«-i>e»a-«>« = W6[?(fl),I2(i3)]fl
and therefore the theorem is proved. D
From Theorem 7.1 with the results of Chapter 1, Section 11.5, we
deduce
Corollary 7.1. // integer s > 0, we have
[Es{Q),L2{Q)}e = {u\ue &'(Q), Q5il~9> u e H^-'^Q)}
for 0 < 6 < 1, with s(l — 6) + \ 4= integer.
Using Proposition 7.1, we obtain
Corollary 7.2. // integer s > 0 0w^ s(l — 9) zs aw integer, we have
[&(Q),L>(Q)}e = 5«i-<»(Q).
Using the reiteration theorem (Chapter 1, Section 6.1) we also obtain
Corollary 7.3. // integer s > 0, we have
[ES{Q),L2(Q)]9 = S^-^iQ) V0, 0 < 6 < 1.
Indeed, let k < s{\ — 6) < k + 1, integer &, then by Definition 6.2,
we have
SS^-9)(Q) = [£*+1 (£),£*(£)V> with 0' = 1 - [s(\ - 0) - k].

7.2 Use of Interpolation; First Results 185
But from Corollary 7.2 and the reiteration theorem (Chapter 1,
Section 6.1) we deduce
[■5*» (Q), E*{Q)}e. = [[£*(£), L2(0)]fc. [S°(Q), L2(fl)],J,.
= [^(i3),I2(i2)]fl.,
with
s — & — 1 s — &
s s
and therefore
6* = [s(\ -0) - ft]——— + {1 - [s(l -0) - ft]}-^— = 0- □
s s
Finally, from Corollaries 7.1 and 7.3, we deduce
Corollary 7.4. For real s > 0, s — \ 4= integer, we have
SS(Q) = {u\ue@'(Q),QsueHs0(Q)}.
7.2 Use of Interpolation; First Results
We have already applied the theory of interpolation in Section 5.4
to study the problem (5.1) in the spaces HS(Q) with real s > 2m and
in Remark 6.5. We shall use it again now, in an essential way, for the
case real s, 0 < s < 2m, which is still missing. We assume that
hypotheses (i), (ii), (iii) of Section 5.1 are satisfied.
From Theorems 5.4 (s = 2m) and 6.6 (s = 0), we deduce that the
operator SP = {A; B0, . . ., Bm_l} is an (algebraic and topological)
isomorphism of
(7.15) H2m(Q)IN onto \l2{Q) x n H2m-mJ~1/2(r); N*,A
and of
(7.16) D°A(Q)IN onto \s-2m(Q) x ]J H~mJ-1/2{r); N*,A .
By interpolation, it follows that
(7.17) 9 is an isomorphism of [H2m{Q)jNf DA(Q)jN]e
onto
m-l
\l2 (Q) x n Kim-m*-i/2 cn; n*, A,
!m-l 1
3'2m(Q)x £ H-mJ-ll2(r)\N*,y\
J = 0
m-l
V0, O<0<1.
Now, we have to interpret the interpolation spaces we have obtained. D

(7.18)
186 7. Application of Interpolation
First, thanks to Chapter 1, Section 13.4 and to (6.42), we have
IL2 (12) x n H2m~mJ-112 (J1); N*. A ,
\5-2m(Q)x YlH-mJ-1/2(r);N*}A\
I j=o \\e
= \[L2(Q),5-2m(Q)]9 x n H2m^^-mJ-^2(r);N*tA .
Furthermore, thanks to the duality theorem (Chapter 1, Section 6.2)
and to Corollary 7.3, we have
(7.19) [L*(Q),S-»»(Q)]e = {[S^{Q),L^Q)-\1_ey
= (S2m9(Q))' = S~2m9(Q) V0, 0 < 0 < 1.
Also, thanks to Chapter 1, Section 13.4:
(7.20) [H2m (Q)/N, D°A (Q)IN]9 = ([H2™ (Q), D°A (£)]fl)/iV, 0 < 6 < 1.
Therefore, in short
(7.21) 0> is an isomorphism of ([H2m (12), D°(Q)]e)/N onto
s~2m9(Q) x n H2**1-9>-mJ-1t2(r)',N*,r\, o < 0 < i. d
j=o J
The space 1)^(0), 0 < s < 1m
There remains to interpret the space [H2m (12), DA (Q)]e. For this
purpose, we introduce the space
DSA(Q) = {u\ueHs(Q),A ue£s-2m(Q)}, 0 < s <2m,
provided with the norm of the graph
IMId^s) = (IMIfl.ai) + M«l&-2m«i))1/2,
which makes it a Hilbert space.
Theorem 7.2. Under hypotheses (i) #w^ (ii) of Section J, z#£ /mv£
(7.22) [H2m (Q), £>2 (Q)]d = Z^m(1 -fl) (fl) V0, 0 < 6 < 1.
Proof. — 1) First, we note that ^4* is properly elliptic in D] therefore
we may apply Remark 1.3 according to which the Dirichlet problem for
A* satisfies conditions (i), (ii), (iii) of Section 5. Therefore, we may
apply the a priori estimate (5.3) with r = 0 and obtain the fact that
for all functions u e H2m (12) n H™ (Q) and therefore a fortiori for all
ueH20m(Q), we have
Minima,) ^ C{\\A*u\\2L2(Q) + \\u\\2H2m-lw}.

7.3 The Final Results 187
Applying Theorem 16.3 of Chapter 1 we obtain that for all u e Hlm(Q)
we have
\\u\\2H2mm<C'{\\A*u\\2LHQ) + ||«||2W-
The sesquilinear form
7i(u, v) = \ A* u A* v dx + uvdx
defines on H2Qm{Q) a scalar product which is equivalent to the "natural"
scalar product. Let / eH~2m(Q); v -» </, v} is a continuous antilinear
form on H2Qm[Q), therefore it may be expressed by a scalar product
(in the scalar product jz(u,v)):
\<f,v> = n(Gf,v)9 GfeH20m(Q)
(7-23)
I Ge&(H-2m(Q)',H20m(Q)).
Setting Gf = u, (7.23) is equivalent to
(A A* + I)u = f, ueHlm(Q).
2) We shall now use Theorem 14.3 of Chapter 1. Using 1), we see
that this theorem may be applied if we set
X = H2m (Q), Y = L2 (Q) (0 = L2 (Q))
& = L2{Q)} <SI = S~2m(Q) (W = H-2m(Q))
§t = L2{Q)} # = H~2m(Q),
d = A, & = A*(AA* + I)-1
(we note that thanks to 1) (A A* + I)"1, inverse of A A* -f- /, exists
and operates from H~2m(Q) onto H2Qm(Q)),
r = -{A A* + I)-1.
The application of Theorem 14.3 of Chapter 1, under the preceding
conditions, yields Theorem 7.2. D
Remark 7.1. From Theorem 7.2, we deduce that 3f{p) is dense in
D2m^-e)(Q), for it is dense in H2m(Q) (Chapter 1, Section 8.1), and in
D°A{Q) (Theorem 6.4). D
7.3 The Final Results
Applying (7.22), we can now obtain the trace theorem for the space
D5A(Q), 0 < s < 2rn, and the final result on the boundary value
problems.
Theorem 7.3. Under hypothesis (i), (ii) of Section 5 and if the system
{Bj}™^ is normal on r, for all real s such that 0 < s < 2m, the mapping

188 7. Application of Interpolation
u-> B u = {B0u, . . ., Bm_x u) of @(D) into {S)(r)}m extends by
continuity to a continuous linear mapping, still denoted by u -> B u, of
DSA(Q) into
m-l
J = 0
Proof, u -* B u is a continuous linear mapping of H2m (12) into
m-l
l\H2m-mr1/2(r) (Chapter 1, Section 8.2) and of D°A(Q) into
J = 0
m- 1
Y\ H~mJ~1/2(r) (Theorem 6.5); therefore, by interpolation, using
j=o
Theorem 7.2 and the properties of the spaces Hs(r) (Chapter 1, Section7),
m-l
it is a continuous linear mapping of DA(Q) into \\Hs~mrll2(r) for
s=o
real s,0 < s < 2m. Furthermore, this mapping is obtained by extension
by continuity of 2(D) into DSA(Q), for Q)(U) is dense in DSA(Q) (see
Remark 7.1). D
Finally, from (7.21) and Theorem 7.2, we deduce
Theorem 7.4. Under hypotheses (i), (ii), (iii) of Section 5, the operator
SP = {A; B0, . . ., Bm_!} is an algebraic and topological isomorphism of
DA{Q)IN onto
m-l
Es~2m(Q) x [] Hs-mJ-1/2(r))N*,f
for all real s such that 0 < s < 2m. Q
Again we note that, in DSA(Q), SP is an operator with index
%(0>) = dimiV - dimiV*.
Remark 7.2. Thus having concluded the study of boundary value
problem (5.1) in the spaces HS(Q) with arbitrary real s, we believe that
it might be useful to the reader if we summarize the various results in
one statement.
Under hypotheses (i), (ii) and (iii) of Section 5.1 (which, as we recall,
define the "regular elliptic problems"), we consider the boundary value
problem
(7.24) A u = / in Q
(7.25) BjU = gj on T, j = 0, 1, . . ., m - 1.
Let s be an arbitrary real number] let
feHs~2m(Q) if s^2m, or feSs~2m(Q) if s<2m,
gjeHs-mJ~1/2(r), / = 0, l,...,m - 1.

7.3 The Final Results 189
Then, these exists u e HS(Q), "solution" of (7.24), (7.25), if and only if
(7.26) </, v> + X <gj, T^> = 0, V*; e N*
j=o
where
N* = {v\veS(D)9 Cjv = 0,; = 0,...,m - 1, 4% = 0}.
S^c/& a "solution" u is determined up to addition of a function of N
(i.e. if weN, u + w is again a "solution"), where
N = {w\ w e^(D)tBjW = 0,/ = 0, . . ., m - l,Aw = 0}.
Fhe term "solution" of (7.24), (7.25) is to be understood in the
following sense:
a) (7.24) is verified in the sense of distributions on Q\
b) (7.25) is verified in the sense of the various trace theorems (for
s^2w, Chapter 1, Section 9.2, for 0 < s < 2m, Chapter 2, Section 7,
for s < 0, Chapter 2, Section 6), which all depend on an extension by
continuity (and therefore "natural")1 of the classical trace operator on r
for sufficiently regular functions in Q. Finally, the solutions depend
continuously on the data in the sense of the following estimates:
( m~l i
" ||fJs-my-l/2Gr)J
(7.27)
(7.28)
inf || m + z||h.(0) ^ C •
zeN
M«llfl.-2-.(|,) + Z H^Wllfl
[ J = 0
if s ^ 2m
inf ||« + z\\Hs(Q) ^ C
zeN
m-l
M«lls.-2«(fl) + 111^*111,.
j = 0
-m;-l/2(rH
if s < 2m.
Remark 7.3. As we have already pointed out, the space Ss~2m(Q)
intervening in Sections 6 and 7 (when s < 2m) is not optimal. We shall
give an example for which the space Ss~2m(Q) may be improved upon
(and for which we can obtain the optimal space).
We go back to Section 6.1 and, in order to simplify, we assume that
N = N* = {0}; then, with the definitions introduced in Section 6.1,
we have
(7.29) A (resp. A*) is an isomorphism of H2Bm(Q) (resp. H2cm(Q))
onto H°(Q).
By transposition (this is exactly what we have done in Section 6.1,
see Proposition 6.1 for r = 0):
(7.30) (A*)* is an isomorphism of H°(Q) onto (H2cm(Q))';
1 See also Section 8.1 below.

190 7. Application of Interpolation
Thanks to the formula (2.3) we immediately verify that
(A*)*(p = A(p, V(peH2Bm(Q).
Therefore (^4*)* is an extension of A and we set
{A*)* = A.
Then, interpolating between (7.29) and (7.30), we have:
A is an isomorphism of [Hlm(Q), H°(Q)]d onto
[H0(Q)(H2cm(Q))']d, 0<6< 1.
The second space coincides (duality theorem, Chapter 1, Section 6.2)
with ([Hlm(Q)}H°(Q)]1_ey. But, according to Theorem 14.4 of
Chapter 4 (the proof of which does not use this remark!), we have
(7.32) [H2cm(Q), HQ(Q)]^d = H2m{Q)
if
2m — jLtj - 1 (= order of Cj) > 2m6 — %.
On the other hand, according to P. Grisvard [8] (see also Remark 14.5,
Chapter 4, Volume 2), we have, if
mj < 2m(l — 6) — \, j = 0, . . ., m — 1,
[H2Bm(Q),H°(Q)]d = H2B"«1-d)(Q)
= {v\veH2m(1-d)(Q),BjV = Oon T, 0 ^ / g m - 1}.
Consequently:
(7.33) A is an isomorphism of H2Bm(1~d){Q) onto (H2md{Q))'
if 2m 6 < \.
But if 2m d < \, we have (Chapter 1, 'Section 11):
(H2m(Q))' =H-2me(Q).
Therefore:
Theorem 7.5. Assume that hypotheses (i), (ii), (iii) are satisfied and
that (7.29) holds. Let 0 < s < \. Then A is an isomorphism of EBn~s{Q)
onto H~S{Q).
We combine this result with Remark 7.2 to obtain: in (7.24), (7.25),
assume that
/ e H-S{Q), gj e H2m's'mJ' ^ (r), 0 < s < ±.
Then the solution u e H2m~s(Q) (compare also with Section 8.3). □
Remark 7.4. The preceding remark also applies to the evolution
equations treated in Chapters 4 and 5, Volume 2; see, for example,
Chapter 4, Remark 15.1. D
(7.31)

8.1 Continuity of Traces on Surfaces Neighbouring J7 191
8. Complements and Generalizations
In this section, we provide some complements to the preceding
theory.
8.1 Continuity of Traces on Surfaces Neighbouring 71
We first reconsider the trace theorems.
Let {re}, 0 ^ q ^ £0 < 1, be a family of surfaces which are parallel
to r and tend towards r when q -» 0.
More precisely, we assume, as hypothesis (i) of Section 5 allows us
to do, that:
there exists a finite family of open sets 0lt . . ., 0N in Rn
such that N
re and r<z{JO,
and that, for each i, there exists an infinitely differentiable
homeomorphism 0t, with non-null jacobian, of 0t onto the
cylinder
fm-1 l
(y.').Iy?< i,-i <*< i
i-i j
satisfying
(a) 6t maps 0t n Q onto Q+ = {(y,t) eQ J > 0},
(b) 6t maps 0,nf onto Q0 = {(y, /) eQ J = 0}
(c) 6t maps 0, n Te onto QQ = {{yj) eQ J = q}
(d) if Ot n Oj 4= 0, then 0t and 9j satisfy the usual
compatibility conditions on Ox n Oj (i.e. there exists an
infinitely differentiable homeomorphism Jij} with positive
jacobian, of 0t (Ot n Oj) onto Oj (Ot n Oj) such that
With the help of {0J, we can define a homeomorphism
(8.2) x->y{x,Q) of Te onto T,
which is infinitely differentiable both ways and where ip and \p~l and
all their derivatives are bounded by constants independent of q (but
dependent on the order of the derivative). Finally, we let QQ be the
open set with boundary re, contained in Q.
We assume, as we are allowed to do, that the coefficients of the
operators {Bj}, {Cj}, {Sj}, {Tj} are defined and infinitely differentiable
not only on r but in D — QQq, so that the systems {Bj}, {Cj}, {Sj},
{Tj} are normal on re for all o with 0 < q ^ q0.
(8.1)

192
8. Complements and Generalizations
We need to consider three cases:
1) s ^ 2m.
We consider the space Hs (Q); if u e Hs (Q), its restriction to Qe
(still denoted by u) belongs to Hs(Qe) and therefore we-may define
BjU\rQ on Te, / = 0, . . ., m - 1 (Chapter 1, Section 9). With the
help of (8.2), we may, by transfer of structure, define the image of
Bj u \rQ on re; we set:
(8.3) Bfu = image under (8.2) of BjU\rQ.
We have Bf u e Hs-mJ~1/2(r) and, using the definition and the
properties of Hs (Q)-spaces (Chapter 1, Sections 7 —9), it is easy to see
that
(8.4) Bfu->BjUin Hs-mJ-^2{r) as q -»0,
/ = 0, . . ., m — 1. D
2) s ^ 0.
Note that in this case (s ^ 0), if u e DSA (Q), its restriction to QQ
does not in general belong to DSA (QQ) (defined in analogous manner as
DSA (O)), for the mapping "restriction " to Qe does not mapSs~2m(Q) into
Ss~ 2m (Qe). Therefore Bj u \Fo can not be defined for arbitrary u e DSA (Q).
This leads to constrain the class of ^'s for which we shall prove (8.4).
For example, we can introduce the spaces
D5A(Q) = {u\ueHs(Q),AueL2(Q)}
and
DsA(Qe) = {u | ueH'(QQ),A ueL2(QQ)}}
provided with the norm of the graph and note that the restriction to
QQ of u e DSA (Q) belongs to DA (Qe). By the same arguments as in
Sections 6.4, 6.5, 6.6, we verify that 2{D) (resp. @(De)) is dense in DSA{Q)
(resp. DA(Qe)) and that, for u e DA(Q) (resp. BA(Qe)), we have a trace
theorem analogous to Theorem 6.5. Therefore, we may define Bju\r
on r and Bj u \Fq on Te for u e DA (Q). Using (8.2), we then define Bf u
by (8.3).
We show that we still have (8.4).
Clearly (8.4) holds if ue@(D). Therefore, it is sufficient to show
that
Bf remains in a bounded set of &(DSA(Q), Hs-mJ-1/2(r)),
j = 0, . . ., m — 1, as q -» 0.
Therefore, we have to show that for fixed q> = (q)0i . . ., <pm-i) in
m-l
n#~s+m'+1/2(r),
J = 0

8.1 Continuity of Traces on Surfaces Neighbouring r 193
we have
m-l
Z (Bfu,^
j = o
^ constant, independent of q.
After a transfer of structure by (8.2), this amounts to showing that
Z <Bj"\r0>9j.Q>
j = o
< constant
when (pe = (q)0>e, . . ., (pm-i,e) belongs to a bounded set of
m-l
n#~s+m'+i/2(/;).
j=o
But according to the definition of Bju\r , we have
Z <BJ u l/v VJ.d> =<U>A* %> ~<Au> «^>,
* = 0
where (u,A*v(Poy denotes the duality between Hs(Qe) and Hos(Qe),
(A u,^^} the scalar pioduct in L2(Qe) and v^ is the right-inverse
of <pQ, analogous to the one in Theorem 6.1 but passing from re to Qe
(therefore
v%eX-°(Qe) = {v\veH2"-{QJ,CJv\rt = 0,j = 0,...,m-\,
A*veHos(Qe)}).
Then it suffices to verify that we may choose v<Pq so that it remains
in a ball of X~s(Qe) whose radius is independent of q, as q -> 0.
But this follows from the proof of Theorem 6.1, since all the "maps"
are realized by functions which together with their derivatives are
bounded in q, thanks to (8.2). □
3) 0 < s < 2m.
We consider Da (12), again defined as for the case s ^ 0. With the
same notation as before, we show (8.4) for 0 < s < 2m, by interpolation
between the cases s = 2m and s = 0 and using Theorem 7.2 of this
chapter and Theorem 5.2 of Chapter 1.
Therefore, we have obtained
Theorem 8.1. Under the hypotheses of the various trace theorems
(Chapter 1, Theorem 9.4 if s ^ m, Chapter 2, Theorem 7.3 if 0 < s < 2m,
Chapter 2, Theorem 6.5 and Remark 6.5 if s ^ 0), if {re} is a family
of parallel surfaces to r which tends to r when q -> 0 (in the sense of
(8.1)), then Bf u-*BjU in Hs~mJ-1/2 (r) as q -> 0, / = 0, . . ., m - 1,
for u e HS(Q) if s ^ 2m and for u e DA(Q) if s < 2m. Q

194
8. Complements and Generalizations
8.2 A Generalization; Application to Dirichlet's Problem
The starting point of Section 4 to obtain the direct a priori estimates
(see (4.39)) was the study of the operator
1 d \ fid
''-W*T* :M*T*
,.„ -B-["-Tli:
We have considered £PV (see (4.2)) as an operator of H2m(R+) into
L2(R+) x Cm.
But suppose that max nij <2m — 1 and set I = 2m — 1 — maxmj. Then
we can see that the theory of Section 4 for the direct estimates may be
developed again if we consider ^ as an operator of H2m~l(R+) into
H~l(R+) x Cm. We only need to use an extension / -» /* of H~l(R+)
into H~l(R) instead of / -» / of L2(R+) into L2 (R) (extension which
exists, see Chapter 1, Section 2) in the proof of Lemma 4.1. We may
then carry on as in Sections 4 and 5 and obtain the following
generalization of formula (5.3) of Theorem 5.1:
Theorem 8.2. Under hypotheses (i), (ii) and (iii) of Section 5 and if
I = 2m — 1 — maxw, > 0,
j
then, for r = — Z, — I + 1, . . ., 0, 1, . . ., we have: if ueH2m~l(Q)
and
m-l
0>u = {A u\ B0u,..., Bm_1 u] eHr(Q) x Y[H2m+r-mJ-lf2(r),
j=o
then ueH2m+r(Q) and
(8.6) || U ||H2m + r(fl) ^ Cr{\\0>U \\Hr(Q)Xmu H2« + r-m/-l/2(r)+ || « || H 2- + r - l(fl)} • D
We point out the consequences of (8.6). Using the same arguments
as in the proof of Theorem 5.2, we see again that ^(r) (notation of
Section 5 extended to the case integer r ^ —I) has kernel N, that Im(^(r))
is closed and that
m-l
Im(^(r)) = Im(^(_f)) n #r(£) x [] H2m+r~mJ-1/2(71),
J = 0
r= -/,..., -1,0,....
It can also be shown that
(8.7) codim Im (^(r)) = codim Im (^(0)), r=-Z,...,-l,
by still using the fact that codim Im(^(0)) is finite and by dual
arguments to those used for Theorem 5.2 to prove (8.7) for r > 0. Therefore
^(r) is an indexed operator and its index % is independent of r\ and
therefore Theorem 5.2 is still valid for r = —I, —Z + 1, . . ., —1. □

8.3 Remarks on the Hypotheses on A and Bj 195
It follows that the theory developed in Sections 5.3 and 5.4 is also
valid under these new conditions: in particular, we have Theorem 5.3
for r = —I, . . ., — 1 and Theorem 5.4 for s ^ 2m — I, with s — \ 4=
4= integer if s < 2m (if s < 2m, the exception occurs because of
Section 12.8 of Chapter 1). D
Starting from this point, it is possible to develop the theory of
transposition and of interpolation as in Sections 6 and 7. We shall not
insist on this point. D
Applied to the Dirichlet problem (for which I = m), the preceding
remarks yield
Theorem 8.3. Under hypotheses (i) and (ii) of Section 5, the operator
{A;y0, . . .,yw_i}
defines an isomorphism of Hs{Q)jN onto
\Hs-2m{Q) x J] H'-'-WWiN*,?]
for all real s ^ m and different from m + \y m + §, . . ., 2m — \. D
Remark 8.1. For the generalizations obtained in this section we have
used the direct estimates for r ^ — I. Q
Remark 8.2. We could make a more precise examination of the
regularity hypotheses on the coefficients of the operators A and Bj and
on the open set Q. We have assumed that the coefficients and the
boundary r are infinitely differentiable so as to be able to develop the
theory for all HS(Q) with arbitrary real s. But the hypotheses could be
specified for each s separately (see, for example, Geymonat [2]).
But, from this point of view, the use of interpolation and transposition
is not very economical, since, for example in interpolation, to obtain a
result in HS(Q) we must start from the corresponding result in Hlsl+1
([s] = greatest integer less than or equal to s). D
8.3 Remarks on the Hypotheses on A and Bj
Let us now take a closer look at the different hypotheses made on
the operators A and Bj.
The four essential steps in Sections 4 and 5 were the following:
I. proof of the direct estimates (5.3);
II. proof of the dual estimates (5.4);
III. proof of the existence of the index of 9 (Theorem 5.2);
IV. proof of the theorem on the alternative (Theorem 5.3).
A first remark is evident: the hypothesis on the normality of the
system ?=o{Bj} was not used in Section 4 and we see immediately that

196
8. Complements and Generalizations
in Section 5 it was used only starting from 5.3, when it was required
to introduce Green's formula (5.24).
Therefore, in order to obtain I, II, III, we only used the hypotheses
that A be properly elliptic, that the B/s cover A and that ms^ 2m — 1.
But it is possible to go further: the method of Peetre [2], which
we have followed, may be used in a more general setting by eliminating
the hypothesis that ms ^ 2m — 1 and obtaining I, II, III starting
from the spaces HS(D) with real s > maxwj + \ (still more generally,
with real s > max/z,- + \, where jUj is the normal order of Bj) and
s =1= integer + \\i s <2m. A partial example was given in Section 8.2;
for the general case, we refer the reader to the original work of Peetre [2].
Thus, we see that the hypothesis on the normality of {Bj} and the
hypothesis on the order of Bj are essentially connected to the use of
Green s formula] therefore, they play an essential role in IV and in the
use of interpolation and transposition in Sections 6 and 7. But for
I, II, III, only the proper ellipticity of A and the condition that {Bj}
covers A are required.
We note that these last hypotheses are necessary, more precisely, it
can be shown that the proper ellipticity of A and the condition that
{Bj} covers A are necessary conditions to obtain I (by reduction to the
case R+ and use of Remark 4.2, see Agmon-Douglis-Nirenberg [1]). D
8.4 The Realization of A in L2 (ii)
In Section 6, we have already considered the differential operators A
and A* as (bounded) operators in certain Hilbert spaces. It is also
important to consider them as unbounded operators in L2(Q).
We shall denote by A2 the unbounded operator in L2 (Q) defined by
D(A2) = H2Bm(D) = {u\ueH2m(Q),BjU = 0, / = 0, . . ., m - 1},
A2u = A(x, D) u for ueD(A2).
Similarly, we denote by A* the operator defined by
D(A*) = H*m(Q) = {u\ueH2m(Q),CjU = 0, / = 0, . . ., m - 1},
A*u = A*(x,D)u for ueD{A*).
A2 (A*) is called the realization of A (A*) in L2 (Q) under the boundary
conditions Bj u = 0 (Cj u = 0), / = 0,. . ., m — 1. In the sequel,
we shall sometimes denote the operator A2{A%) by ^4(^4*) if there is
no danger of confusion.
Theorem 8.4. Under hypotheses (i), (ii), (iii) of Section 5, A* is the
adjoint of the operator A2, in the sense of unbounded operators in L2 (D),
i.e. At = (A2)*.

8.4 The Realization of A in L2{Q)
197
Proof. We apply Theorem 5.3 to problems {A,B} and {A*,C};
there results that A2 and A* are closed operators in L2(Q), with closed
image,
Ker^2 = N, KerA* = N*, Im(A2) = {L2(Q); N*}
(orthogonal subspace of iV* in L2(Q)) and Im(Ai) = {L2(Q))N}.
It is also obvious that D(A*) and D(A2) are dense in L2(Q).
But the adjoint (A2)* of A2 in L2(Q) is also closed, has domain
D((A2)*) dense in Z,2(i2) and closed image; it is defined by
(A2u,v) = {u} (A2)*v) VueD(A2), veD((A2)*).
Therefore, we have
Ker(^42)* = orthogonal subspace of Im(A2) in L2 (Q)
and therefore
(8.8) Ker(^2)* = N* = KerA*2.
We also have Im((^42)*) = orthogonal subspace of Ker^42
= {L2(Q);N} in L2(Q)] therefore
(8.9) Im((A2)*) = Im(A*).
Applying Green's formula (5.14) we see that A* c (A2)*. Therefore,
it is sufficient to verify that D((A2)*) c D(A%). Let u e D({A2)*): thus
/ = {A2)* u e Im((A2)*) = Im(A*); consequently, there exists veD(A*)
such that A\v = / and therefore veD(A%) c D({A2)*) and (A2)*v = f.
It follows that u — v e Ker(^42)* = Ker^* <= D(A*)m, from which we
deduce that u = (u — v) + v e D(A2). U
Of course, we may also consider A as an unbounded operator in
Hr (Q) with real r ^ 0; more precisely, we consider in Hr (Q) the operator
(realization of A in Hr(Q)) A2r defined by
D(A2tr) =H2Bm+r(Q) = {u\ueH2m+r(Q),BjU = 0,j = 0, ...,m- 1}
A2r u = A(x, D) u for ueD(A2r).
Similarly, we define A*tf.
Thanks to Theorems 5.3 and 5.4, it is easy to see that A2r is the
restriction of A2 to Hr(Q) and that Ker^42r = Ker^42 = N and
Im(^2>r) = {H'(Q)',N*}. Q
Remark 8.3. From what we have seen for A2 there also results that A2
is an indexed operator in L2(Q) and that
(8.10) x(Ai) = X{&) = dimN - dimiV*,
where 9 is the operator 9 — {A; B0, . . ., Bm_1} defined in
Section 5. D

198
8. Complements and Generalizations
8.5 Some Remarks on the Index of 9
We shall now make some remarks concerning the index of the
operator SP = {A; B0, . . ., Bm_1} (or of the operator A2, thanks to (8.10)).
We have seen in Sections 5,6,7 that SP, as an operator in Hs (Q), if
s ^ 2m, or in DSA(Q), if s < 2m, always admits an index %(&) which
is independent of s and which is given by
(8.11) x{?) = dimN - dimiV*. D
Remark 8.4. It is known (see, for example, Kato [5]) that the index
of an operator between Hilbert spaces (for example!) does not change
if we add a compact operator to the operator. Therefore, according to
the compactness theorem of Section 16 of Chapter 1, if we consider the
operator
J = {A + Q; B0 + F0, . . .,Bm_t + Fm_t}, with Q (resp. Fj)
a linear differential operator of order <2m (resp. < mj) with coefficients
in 3(D) (resp. 0(jT)), we have
therefore the index of SP does not change if we add operators of smaller
order to A and Bj. Q
Remark 8.5. It is also interesting to know when
(8.12) x(0>) =0, i.e. dimiV = dimiV*
(then 3P is sometimes called a Fredholm operator).
A sufficient condition is
(8.13) Hlm(Q) = Hlm(Q) (i.e.D(A2) = Z)(ISJ)
(here E denotes the set of conjugate complex functions of a function
space E).
Indeed, in this case we can consider the differential operators A*
and Cj (j = 0, . . ., m — 1) as being deduced from A* and Cj by
replacing the coefficients with their complex conjugates. Let A* be the
realization of A* in L2(Q) under the boundary conditions CjU = 0,
/ = 0, . . ., m - 1; we see that D(A$) = D(A2). But A - A* is an
operator of order ^2w — 1, therefore A2 — A2 is a compact operator
of D(A2) into L2 (Q); it follows that X(A2)=X(I$). But x(A*2) =X(A*2)
and, thanks to Theorem 8.4,
X(A*2) = -X(A2).
Therefore
X(A2) = -X(A2)=X^)=0-

8.6 Uniqueness and Surjectivity Theorems
199
In particular, condition (8.13) is satisfied for the Dirichlet problem,
since then
H2Bm(Q) = H2cm(Q) = H2m(Q) n ff?(fl).
Therefore
I for every properly elliptic operator A, the index of the Dirichlet
problem is zero. D
We also note that, because of Remark 8.3, to have %(&*) = 0, it
is sufficient to find Q, F0, . . ., Fm_1 under the conditions of Remark 8.3,
such that %(££) = 0. In particular it suffices to find AeC such that
X(A2 + XI) = 0, where I is the identity in L2{Q). Q
8.6 Uniqueness and Surjectivity Theorems
Two other important questions are:
1) when is dimiV = 0?
In this case, there is a unique solution to boundary value problem
{A,B};
2) when is dimiV* = 0?
In this case, there exists a solution of problem {A, B} for all given /
and gj [surjectivity of dP).
The two questions are evidently of the same nature: surjectivity
for {A, B) is uniqueness for the adjoint problem {^4*, C}. □
In most applications these questions are asked in a little more general
way, that is: to give sufficient conditions for uniqueness (or surjectivity)
for problem {A + A /, B) for at least one complex number A.
Of course, we shall have uniqueness for {^4 + XI,B) if we can
show an inequality of the type
(8.15) \\u\\H2m(Q) £c\\Au + Xu\\L2(Q), VueH2Bm(Q).
In Chapter 4, Section 4, we shall prove a sufficient condition for
(8.15) to be satisfied for all A with Re A > £0 (£0 = suitable real number);
it is the following: in Q x Ry, consider the operator
n n
T'T
A9'=A(x,Dx) +e,9(-l)wD;w V0 e
with the boundary conditions
Bj(x, Dx) given on T x Ry, / = 0, . . ., m - 1;
and assume that for all 6 e
n n
and that the system {Bj}J'q covers A9 on fxRy.
, Ae is properly elliptic in D x R.
y

200 9. Variational Theory of Boundary Value Problems
Under this condition, we have uniqueness for {A + XI, B} for all A
with Re A > £0 • For example, the condition is satisfied for the Dirichlet
problem, if A is strongly elliptic in D.
Other sufficient conditions are provided by variational theory (co-
erciveness conditions, see Section 9).
9. Variational Theory of Boundary Value Problems
In this section we shall give the main outline of the "variational''
theory of boundary value problems and compare the results with those
obtained in the preceding sections.
9.1 Variational Problems
The linear elliptic variational problems correspond to the
minimization of positive definite quadratic forms with homogeneous part
of degree 2 on a Hilbert space — which we shall call V. The most
classical example is the Dirichlet problem for the Laplacian: on the space
V = Hq (Q), we consider the quadratic form
Q(v) = a(v,v) -2Jfvdx,
where
" f du dv
a(u,v) = X — -r—dx
1 = 1 J dXi dxt
and where / is given in H~1(Q)l fvdx = </,£>]; then the inf. of
Q (v), as v describes V, is reached by the unique element u of V satisfying
(9.1) a{utv)=(ftv") VveV
or by the unique element of V satisfying
—Au = / in Q,
which may also be written:
[ ueHx(Q)9
(9.2) \-Au = f in Q
\ u = 0 on f.
Formulation (9.1) leads to the "abstract" problem:
let u, v -> a (u, v) be a continuous sesquilinear form on V x V and
let / e V = antidual of V((f, v) denoting the value of / at v)\ we seek
u e V, solution of
(9.3) a(u,v) = (f,v) VveV. Q

9.1 Variational Problems 201
Remark 9.1. In (9.3), we do not necessarily assume that a(u,v)
= a(u,v); problem (9.3) therefore no longer corresponds to a problem
of minimizing a quadratic form. Nevertheless, because of its close
relation to the problems of the calculus of variations, problem (9.3)
is called a "variational problem". D
Remark 9.2. a(u,v) may be represented in the form
(9.4) a(u,v) = [Au}v)} Ae&{V)Vf)
and, for the moment, A is arbitrary in if (V; V). Then (9.3) is equivalent
to the most general linear equation
(9.5) A u = /
for A e£?{V\ V). But we shall give convenient sufficient conditions
for (9.5) to admit a unique solution. D
If a(u,v) = a(u,v), then we associate the form
Q(v) =a(v,v) -2(f,v)
to (9.3) and we are led to assume a(v, v) to be positive definite on V.
In the non-symmetric case, we consider the real part of a (u, v) and we
are led to the following definition:
Definition 9.1. The sesquilinear form a(u,v) is said to be V-elliptic if
(9.6) Rea(v,v) ^ oc \\v\\2 Vv e V, <x<0,
|| v || = norm of v in V.
Then, we have
Theorem 9.1. Under hypothesis (9.6), problem (9.3) admits a unique
solution, the mapping f -> u being continuous from V -> V (or: A is an
isomorphism of V onto V).
Proof. — 1) Denote by || ||* the norm (dual of || ||) in V. We have
*M|2 ^Rea(v,v) = Re(,4 ^) ^ M v||« ||v||,
therefore
(9.7) \\Av\\^oc\\v\\ V^eF.
2) We introduce the adjoint form a*(u, v) by
(9.8) a*(u,v) = a{u,v) Vu,veV.
a* (u, v) may be represented by
(9.9) a*(u,v) = (A*u,v), A*e&(V',V)
and we immediately verify that A* is the adjoint of A.

202 9. Variational Theory of Boundary Value Problems
Since Rea*(v,v) = Rea(v,v) we have, in the same fashion as
for (9.7):
(9.10) \\A*v\\*^<x\\v\\ VveV.
3) The theorem follows from (9.7) and (9.10) (the operator A is
one-to-one and the image under it is dense and closed in V). □
Remark 9.3. The preceding result is valid, with the same proof,
under the hypothesis
(9.11) \a(v,v)\ ^<x\\v\\2, oc>0, Vv e V. Q
Remark 9.4. For many applications — and in particular, as we
shall see in the following chapters, for all the theory of evolution equations
— we must consider, not the problem (we use the notation of Sections 1
to 8):
Au = f, BjU = 0, 0^/^w-l,
but the problem
Au + Au = f, BjU = 0, 0^/gw-l,
where X is a complex parameter. This leads to a variational formulation
in which two Hilbert spaces intervene, instead of only the space V
(since V is intrinsically connected with V): we consider two Hilbert
spaces V and H, with
(9.12) V c H, V dense in H, the injection of V into H is continuous.
We identify H to its antidual, and if V is the antidual of V, we may
identify H to a subspace of V\ since V is dense in H, therefore
(9.13) V c H cV.
We denote by (, ) the scalar product in H and by | | the norm in
H) if / e V and v e V, their scalar product is also denoted by (/, v)t
which is permissible thanks to the identifications (9.13).
Then, instead of problem (9.3), we consider the problem: find u eV,
solution of
(9.14) a{ut v) + X(u, v) = (/, v) Vv e V,
where X is given in C.
Definition 9.1 is now generalized to:
Definition 9.2. The sesquilinear form a(u,v) is said to be V-coercive
{relative to H) if there exist XQ e R and <x > 0 such that
(9.15) Rea(v,v) + XQ \v\2 ^ * ||v||2 Vv e V.

9.3 A Counter-Example
203
Of course, Theorem 9.1 yields:
Corollary 9.1. // the form a(u,v) is V-coercive (in the sense of (9.15)),
then problem (9.14) admits a unique solution for all AeC satisfying
(9.16) ReA^V D
9.2 The Problem
We take up problem (5.1) again, in a formal way (i.e. without worrying
about the data and solution spaces).
If (p is a "function" such that
Bj<p = gj, 0 ^ / ^ m — 1,
then
u — (p = w
satisfies
A w = f — A <p
BjW = 0, Og/^m-1.
Therefore, at least in a formal way, we are led back (changing the
notation) to the problem
A u = f
BjU = 0, Ogjgw-1,
to which we also associate (Remark 9.4) the problems
I A u + A w = /
BjU = 0, 0 ^ / ^ w - 1.
The question is: which "regular elliptic" problems (in the sense of
Sections 1 to 8) of type (9.17) (resp. (9.18)) belong to the class of V-elliptic
(resp. V-coercive) variational problems?
We start with a counter-example which is due to Seeley [3].
9.3 A Counter-Example
In R2, consider (we use polar coordinates (r,0)):
Q = {(r,0) \tz <r < In).
Take the elliptic operator
<9j9) ^=-(e^)2-eM(i+^)
and associate problem (9.18) to it, with the Dirichlet condition:
(9.19') B0 = trace on T ( = boundary of Q).
(9.17)

204 9. Variational Theory of Boundary Value Problems
For the choice (9.19), (9.19)', problem (9.18) is never V-coercive (and
no matter what the possible choice of V is). Indeed, we shall verify
that, no matter what X is, problem
(9.20) (A + X)u = 0, u\r = 0
admits non-null solutions (which, according to Corollary 9.1, would be
impossible if the problem was F-coercive). Indeed, if ju e C satisfies
fx2 = A, the functions
u = sinr cos(/z e~i0) and u = sinr sin(jn e~i0) [X #= 0)
and
u = sinr and u = e~i9 sinr (A = 0)
are solutions of (9.20). D
9.4 Variational Formulation and Green's Formula
Let A be defined by (1.5) and a(u,v) by (2.18). We consider
problem (9.17) and apply Green's formula (2.19).
Among the B/s, choose those which are of order mj < m\ we may
always assume, with an eventual permutation of indices, that they are:
B0, . . ., Bp_!, 0 ^ p S m — 1 {P = 0 means that all B/s are of order
^.m). There certainly exist "boundary" operators B'p, . . ., B'm-X, with
infinitely differentiable coefficients on r, such that the system
{B0, . . ., Bp_lf B'p, . . ., B'm-i} is a Dirichlet system of order m on r.
We can apply Green's formula (2.19), taking {B0, . . ., Bp_lt
B'p, . . ., B'm-i} for the system {Fj}. Formally, we can say that, if v
satisfies
(9.21) B0v = 0,...,Bp_lv =0,
then for all u, we have
(9.22) a{u,v)= (Au)vdx - £ OjuB'jvda.
q r
It follows that, if ^ satisfies
(9.23) a{u,v) = [fvdx, Vv satisfying (9.21),
then
(9.24) il u = / in £
and
m— 1 /»
(9.25) £ #|WB><*<T = 0, Vt; satisfying (9.21).
j=p J

9.4 Variational Formulation and Green's Formula 205
Since v is arbitrary and since {B0, . . ., Bp_ t, Bp, . . ., B'm-1} is a Dirichlet
system of order m on r, it follows that (see Lemma 2.2):
(9.26) 0^=0, j = p,...,m-\.
Conversely, if u satisfies (9.24) and (9.26), then it follows from (9.22)
that u must also satisfy (9.23).
Thus, we see how we can formally put problem (9.1) into a variational
formulation (we do not yet worry about the coerciveness): assume that
we can choose B'p, . . ., B'm-i so that
(9.27) &j = Bj. j = p,...,m-\.
Then (9.17) is equivalent to (9.3), taking for V the space of v's in Hm(Q)
which satisfy (9.21). □
Remark 9.5. In the variational formulation we must therefore divide
the boundary conditions into two groups:
BjU = 0, / = 0, . . ., p — 1, with mj < m,
which are sometimes called stable conditions, and
BjU = 0, j = p,...,m—\, with m5^.m,
which are called natural or transversality conditions. D
Remark 9.6. Therefore, we can reduce problem (9.17) to the variational
formulation if we can choose B'p, . . ., Brm-i so that (9.27) is valid. But
this imposes restrictions. Indeed, in Green's formula (2.19), A being
elliptic, we have: order of 0j = 2m — 1 — order of Fj. Therefore, if
ftp, • • •, /Wm-i are numbers between 0 and m — 1 so that mx, . . ., mp_x,
ju>p> • • •, /Wm- i yield all the numbers 0, 1, . . ., m — 1 (in arbitrary
order), then /Uj is the order of Bj and therefore the order of BJt with
j = p, . . .,m — 1, is given by 2m — 1 — jujt that is
(9.28) mj = 2m -- \ - fjij, j = p, . . .,m - 1.
Therefore, we have found a restriction on the order of the natural conditions
(in the sense of Remark 9.5). This condition is not necessarily satisfied
for a regular elliptic problem (i.e. satisfying conditions (i), (ii), (iii) of
Section 5). □
Here is an example. Let
(9.29) Au =A2u + u = V —T[—r ) + u
ij=i dxt \dxj J
with the boundary conditions given by the boundary operators
dAu
(9.30) B0u = u, Blu = (v = normal to r, directed
3 v
towards the interior).

206 9. Variational Theory of Boundary Value Problems
Then, we have m = 2, p = 1, m0 = 0, [ix = 1, mx =3 and therefore
(9.28) is not satisfied for / = 1.
Nevertheless, problem (9.29) — (9.30) satisfies conditions (i), (ii),
(iii) of Section 5. Indeed, we can reduce it to the case of the half-space
Q = Rn+. In the notation of Section4.1, the polynomial A0(t],r)
becomes
A0(r]>r) = (r]2 + r2)2;
thereforer+ = irj, double root, M+ (rj}r) = (r — irj)2. The polynomials
corresponding to B0 and B1 are
B0(rj,r) = 1, fiifo.r) =r(rj2 + r2).
Condition I) of Section 4 is satisfied; to verify II) it suffices to verify
that if a + br(rj2 + r2) is divisible by (r — irj)2, then a = b = 0,
which is immediate.
Finally, B0 and Bx form a normal system on r. Therefore problem
(9.29) —(9.30) fits the theory of Sections 1— 7, but not the variational
formulation. □
Remark 9.7. In fact, all the considerations of Section 9.4 depend on
a choice] starting from A u = /, one multiplies by v and integrates
on Q, then one applies integration by parts formulas; in other words
one takes the scalar product in L2(Q). But, more generally, we can
"replace'' the equation A u = f by the equation
(9.31) f Au/lvdx = jfA~vdx,
where the operator A is at our disposal and is to be chosen in a suitable
way.
This will eventually give us a space V depending on the choice of A.
So that what we have in fact shown in Remark 9.6 is that, for the
choice A = identity, there exist regular elliptic problems which do not
fit the variational setting. But we shall see that, for example for (9.29),
(9.30), we can choose A in such a way that the problem fits the
variational, F-elliptic setting. □
State of the problem
We started by examining when a regular elliptic problem can be
put into variational form.
Now, conversely, we shall see which boundary value problems
correspond to variational problems (Section 9.5). Then, we shall briefly
study (Section 9.6) the coerciveness of variational problems.

9.5 "Concrete" Variational Problems
207
9.5 "Concrete" Variational Problems
Let Q be an arbitrary bounded open set in R" and
,Au= £ (-l)MD>(apq(x)D«u),
(9.32) | l*l-l«l§m
\a„eL*>(Q).
In the notation of Section 9.1, we choose 7 and H in the following
manner:
H = L2(Q),
V = closed subspace of Hm(Q) such that H™(Q) e 7,
with continuous injection.
We also consider a Hilbert space K (in order to stay in the hilbertian
setting, but we could also consider a non-hilbertisable topological vector
space) which is a normal space of distributions on Q and such that 7 c K,
with continuous injection (for example K = L2(Q)). Let K' be the
dual of K; note that K' is a space of distributions on Q.
We consider the continuous sesquilinear form on 7:
(9.33) a(u,v) = £ LMD«^^,
|p|,|fl|^m J
The variational problem is the following:
I with given f e K'' z#£ s££& ^ e 7 s^c/& tfto
«(«, v) = </, v> Vv e 7,
where the brackets denote the duality between K and K'.
If the form a(u,v) is 7-elliptic, then we may apply Theorem 9.1,
since </, v} is a continuous antilinear form on 7, according to the
hypotheses on K. D
Let us give an interpretation of the problem.
First, since 2{Q) c 7, we deduce from (9.34) that
(9.35) Au = f,
in the sense of distributions on Q.
As far as the "boundary conditions" for u are concerned, they are
partly contained in the condition u e V (stable conditions) and partly in
equation (9.34) (natural conditions).
If 7 is determined by differential conditions such as (9.21), we may
also interpret the natural conditions by using Green's formula (2.19)
(see Section 9.4); but this is purely formal if we do not have regularity
conditions on the data and on the solution u which enable us to justify

208 9. Variational Theory of Boundary Value Problems
the use of formula (2.19). Therefore, in general, we may say that the
natural conditions are taken'in the sense of equation (9.34).
Usual examples:
1)7 = Hq(Q) . Then we have Dirichlet's problem; all the boundary
conditions are stable: we have ys u = 0, / = 0, . . ., m — 1 (and the
traces yj u on T have meaning if Q is sufficiently regular).
In this case, we may take K = Hg (Q) and therefore K' = H~m (Q). D
2) V = Hm (Q). There are no stable boundary conditions. Formally,
using (2.19), we obtain the natural conditions
0jU = 0, / = 0, . . ., m - 1.
This problem is called the Neumann problem (with respect to Green's
formula (2.19)).
Note that we may have different Neumann problems relative to a
given operator A. Indeed, if we decompose the operator A in different
ways with respect to the elementary operators Dp, we may associate
different forms a(u,v) and therefore different Green's formulas (2.19)
to it; for example, we may decompose the Laplace operator A in the
usual way (take n = 2 for the sake of simplicity):
d I du \ d I du
Au = -— -— +
or
Au= — [— + __ _ +
dxx \dxl ] dx2 \dx2
d I du \ d I du \ d I du
c-
dxx \dx1J dx2 \dx2/ dxt \ dx2j dx2 \ dxx
dc du dc du
dx2 d%i dxx dx2
where c is a real function belonging to C1 (D). Then we have the two
Green's formulas
f / du dv du dv \ C C du
1 dx = —\Auvdx— vda,
J \dxt dxx dx2 dx2j J J dv
q q r
du dv du dv du dv du dv dc du
H h c c 1 v —
dxx dxx dx2 dx2 dxx dx2 dx2 dxx dx2 dxx
dc du \ C CI du du\
v\ dx = — Auvdx — h c \vda,
3*i dx2 ] J J \dv do J
q r
where and denote the derivative along the interior normal
dv da
to r and along the tangent to r, respectively. Corresponding to these

9.6 Coercive Forms and Problems
209
two decompositions we have the two Neumann problems:
du
—Au = / in Qt = 0 on r,
dv
du du
-Au = f in Q, \-c = 0 on R
dv da
Usually, the second problem is called the "regular oblique derivative''
problem for A . □
3) Let ro, . . ., rm^1 be m subsets of r (open sets on r) and set
(still with respect to Green's formula (2.19) and assuming Q to be
sufficiently regular)
V = {v\veHm(Q),FjV = 0 on^, / = 0, . . ., m - 1}.
Then, formally, we solve the mixed problem
A u = /,
Fju = 0 on rj9 0jU = O on r - rj9 j=0,...,w-l.
This problem does not belong to the class of regular boundary value
problems in the sense of Section 1. D
4) If Q = Qx u Q2 u E is divided into two parts, Qx and Q2, by
a surface E and if we assume the coefficients of A to be regular in Dx
and D2 separately, we have a transmission problem: u satisfies the
equation A u = / in Qt and Q2 separately, and junction conditions (or
transmission conditions) on E. D
5) We also note that V may be determined by integral conditions
(instead of pointwise); then we have new types of "boundary value
problems" (see also the Appendix to Volume 2). □
9.6 Coercive Forms and Problems
Now, the fundamental problem is to give algebraic necessary and
sufficient conditions for the problem to be coercive. We shall limit
ourselves to the main results.
1) For the classical case of second order elliptic equations with real
coefficients, it is easy to see that the ellipticity of A is sufficient for the
coerciveness of a(u, v) onH1 (Q) and therefore also on each subspace V\
therefore, in particular, the problems of Dirichlet, of Neumann, of
mixed type and of transmission for second order equations can be solved.
2) For equations of higher order than the second, the first basic
result is due to Garding [1] and Vishik [1] for the Dirichlet problem:

210 9. Variational Theory of Boundary Value Problems
Theorem 9.2. Let Q be a bounded open set in R" and A be given by
(9.23); Assume that apqeC°(D) if \p\ = \q\ = m and that A is uniformly
strongly elliptic in D. Then a(uyv) is coercive on H™(Q) (i.e. the Dirichlet
problem is coercive).
For the proof, which by now is classical, we refer the reader to
Yosida [2], for example. D
3) The coerciveness of the form a(u, v), given by (9.33), on spaces
V which are determined by a number p <^ m of differential conditions
on r of order < m:
V = {v\veHm(Q),BjV = 0, / = 0, . . ., m - 1}
has been studied by many authors, following Aronszajn [1]. One may
use methods analogous to those we have described in Sections 4 and 5
for the direct a priori estimates.
The most general theorem (at least from the point of view of the
operators^! and BJ), which is due to Agmon [2], is the following:
Theorem 9.3. Assume that
a) Q is a bounded open set in R", of class Cm, A(x, D) is given by (9.32)
with apqeC°(D) if \p\ = \q\ = m;
b) Bj, j = 0, . . ., p — 1, are p differential boundary operators
Bj(x,D)= £ bjk(x)D\
with 0 <; mj ^ m - 1, bjh e Cm-mj(r);
c) Re X *«(*) £p+q > ° v% e £, vi e R« - {0}
M=|«| = m
(therefore A is strongly elliptic in D);
d) for alt x e r, all ij e Rn — {0} and tangent toTatx, all £' e R" — {0}
and normal to T at x:
Re f £ aM (x) U + ?- -%-)'v (t) U + p - ^-)9v (t)dt>0
o
for every function v(t) e^(Rt) and =£ 0, satisfying the ordinary
differential equation
loUf + P
and the conditions
1 d \
Bj,o[x.e + e— — )v(t)
1 d\
i dt) K
= 0, / = 0 p-l,
where A0, BJ0 are the characteristic forms of A = \{A + A*) and Bt
(see Section 4.5).

9.6 Coercive Forms and Problems 211
Then the form a(uyv) given by (9.33) is coercive on the space V:
V = {u\ueHm{Q)yBjU = 0, / = 0, . . ., p - 1}. □
The hypotheses on the open set Q can be considerably weakened in case
the form a{uyv) is formally positive (i.e. of the type
a(u,v) = \ £ Ak(xy D) u Ak(x, D) v dx
Q
with Ak a linear differential operator of order ^ m) and V = Hm(Q)
(see K.T.Smith [2]). □
Remark 9.8. Here is an example of a non-coercive variational problem.
Let A u be given by
(9.36) A2u = £ ttttI
i,j=i dxt \dxj J
The form a(u,v) is
" f d2u Wv
i.j=ij 8xt dxj
and problem (9.34), choosing V = H2{Q), is formally equivalent to
the boundary value problem
A2u = / in Q, B0u = 0, Bx u = 0 on T,
with
dAu
(9.37) B0u=Au, Blu = .
8v
The form a(u,v) is not coercive on i/2(Q); it is sufficient to note that
a(u,v) vanishes if u = v = an arbitrary harmonic function in Q. □
Remark 9.9. We can add a suitable "tangential'' form to the form
a(u,v), for example of the type
m- 1
x<^,iv^>,
j = 0
where ^jE^(Hm{Q)\H~m+mJ+ll2{r)) and F,- is the operator of
formula (2.19), Mj being the order of Fjt and impose the coerciveness
condition on the new form
m-l
a{ufv) + £ {^jU.TJv}.
j = o
In this way, we obtain regular oblique derivative problems.
Similarly, we can add a coercive form on r, of higher order than a,
to a{u,v)) thus, we obtain variational boundary value problems for
which the boundary conditions contain derivatives of higher order
than A. D

212 9. Variational Theory of Boundary Value Problems
9.7 Regularity of Solutions
Another very important question concerning the preceding theory
is the regularity of the solution. The solution of problem (9.34) belongs
to V and therefore to Hm(Q); we would like to know the regularity
properties of this solution as a function of the regularity properties of
the data of the problem (open set Q, space V, coefficients of A, /); this
is a problem of a priori estimates.
If Q and the coefficients of A are sufficiently regular and if problem
(9.34) is equivalent to a regular elliptic problem (in the sense of Sections 1
and 8), we realize that the regularity of the solution can be obtained by
the same methods as were used in Sections 4 and 5 and that the results
are of the same type; in particular, we have: if / e Hr(Q), r ^ 0, then
ueH2m+r(Q).
But (9.34) also contains non-regular elliptic problems, as we have
already seen; and here many different situations exist.
(i) There may be a local regularity of the same type as for regular
elliptic problems: such is the case, for example, when the boundary
conditions determined by (9.34) are equivalent to regular elliptic
conditions, in the neighbourhood of a point of r.
Example 3 of Section 9.5 [mixed problems) admits a local regularity
in the neighbourhood of interior points of jT, (with respect to jT). But we
can not obtain a global regularity, of the same type, in Q, that is, if
feHr(Q), we do not in general have u e H2m+r(Q) (see Magenes-
Stampacchia [1] and the bibliography of this work.)
(ii) There are also coercive variational problems which are '' irregular "
everywhere in D. For example, we have seen (Theorem 9.2) that the
Dirichlet problem may be solved in Hm (Q) under very weak hypotheses
on Q and on the coefficients of A; under these hypotheses, the
solution u, even if / is very regular, may not belong to H2m(co), for any
set co contained in D. See the Comments to this chapter for references. □
9.8 Generalizations (I)
Until now, we have considered the differential operator A as a linear
combination of the elementary operators Dp, derivatives of order p. It
seemed natural to take u in Sobolev spaces HS(Q). But, often the
operator A is decomposed into elementary operators different from the
derivatives Dp and then u must be taken in spaces which no longer are
Sobolev spaces. For example, we may write the operator A2u as
" d2 I d2u \ » d2 I d2u\
i,j = i dXidXj \ dXidxjJ itj=1 dxt \ dxj J
rj,fj,eR, with ^+^ = 1,

9.8 Generalizations (I) 213
or as
A(Au).
In the first case, we consider the second derivatives as elementary
operators and it is natural to have u e H2 (Q); in the second case, the
elementary operator is the Laplacian and it is natural to consider the
space of u's eL2(ii) such that Au eL2(Q), which is different from
H2{Q).
Problem
dA u
(9.38) A2u + Au = f in£, Au = 0, = 0 on T,
ov
as we have seen (Remark 9.8), is not coercive on H2 (Q). But obviously,
if we take H = L2 (Q) and V = {v \ v e L2 {Q), A v e L2 (Q)}, provided
with the norm
(\\u\\Lm + \\Au\\2LHO)y<i,
then the form
a(u,v) = Au Av dx + X uvdx
Q Q
is F-elliptic for all X with Re A > 0 and therefore Theorem 9.1 applies.
Therefore for all f e L2 (Q) there exists a unique u eV such that
a(u,v) = fv dx Vv e V.
Q
It follows that A2u + Au = f in the sense of distributions on Q, the
boundary conditions being taken in the formal sense (but note that
w — Au is such that w eL2{Q), Aw eL2(Q) and therefore if Q is
sufficiently regular we may apply the trace Theorem 6.5 and obtain
yoweH-1'2^), yiweH^2{r)). □
Remark 9.10. The variational theory may be applied to non-elliptic
equations (or not even hypoelliptic!); for example, we may study the
problem
d*u
+ X u = / in a square Q of R2, u = 0 on T,
dx2 dy2
in which the operator A is hyperbolic, by taking H = L2 (Q) and V
= closure of 2(Q) in the space
f d2u )
Jf = \u\ueL2{Q), eL2{Q) .
I dxdy )
Another interesting application can be made to a remarkable class
of hypoelliptic operators, which are obtained by using the spaces

214
9. Variational Theory of Boundary Value Problems
Ha,P (Q) and their closed subspaces, where Q = Qxx Qy (Dx: open set
of R" and Qy\ open set of RJT; therefore Q is an open set of Rm+n) and
H^{Q) = {u\ueL2{Q),DpxueL2(Q),\P\ g oc, DqyuEL2(Q) ,\q\ ^ fi}
and as elementary operators the derivatives of the type Dpx and Dq,
with \j>\ ^ oc, \q\ ^ j8. For example, if, in R2, we take
Qx = {x | a < x < b], Qy = [y \ c < y < d],
Q = QxxQy oc = 2, p = 1,
V = closure of 9(Q) in H2>l{Q), H = L2{Q),
d*u d2u
A u =
d#4 dy2
/ Nd2u ~¥l du !h\ \
( and therefore a (u, v) = 1 \dxdy\t
\ J \d#2 d#2 dy dy J
we can solve the problem (of Dirichlet):
A u = / in i2
u = = 0 on the sides x = a and x = 6 of Q,
w = 0 on the sides y = c and y = ^ of i2. D
9.9 Generalizations (II)
We come back to problem (9.29) - (9.30) and Remark 9.7.
Let H = Hh (Q), with the norm
i = i
dv
and
with the norm
V = \v\veH%{Q),
dxt
dAv
dxi
\L2(Q)
1/2
■EL2(Q),i= 1,.
II dv 1
1 d%i
12 1
+
\L2{Q)
\dAv\
\L2(Q)
1/2
We define
" f dAu dAv " f 3w dv
&(»> v) = X i 5—<** + I i J—***-
i = ij d#f dx( i = i J d#f 3^

9.9 Generalizations (II)
215
Then a(u,v) is F-elliptic and Theorem 9.1 applies; therefore, for all
/ e Hq (Q) there exists ueV such that
(9.39)
E
i = l
dAu dAv
dxt dxt
j v f du dv j
i = i J dxt dxt
-I
3/ ~3^
3#f 3#f
<** WeF.
We recall that A is an isomorphism of Ho (12) onto i/~* (12) and that
for (p and ip e Ho (12) we have (Green's) formula
" C d<p dip —
L \—, ;—dx = -<<Mv>>
1 = 1 J fl^i 3 s,
the brackets denoting the duality between H"1 (12) and Hq (12). Then
(9.39) becomes
" f dAu dAv — —
1 = 1 J 3 s, 3%,
= \uAvdx-\fAvdx Vv e7.
But as y describes V, Av describes a space W which, in particular,
contains H1 (12); therefore, it follows that w = A u is a solution of
" f dw dz C
V dx = \(u-f)zdx VzeH1
1 = 1 J 3*, dxt J
(£),
that is of the (variational) Neumann problem
—Aw = u — / in 12,
dw
I
3i>
= 0 on R
But then for u, which belongs to F, we have
(9.41) A2u + u =/ in 12,
(9.42)
dAu
dv
= 0 on T, « = 0 on R
Note that (9.41) is taken in the sense of distributions on 12, and the
boundary conditions (9.42) are taken in the variational sense, if 12 is

216
10. Comments
not regular, and in the concrete sense of trace theorems, if Q is regular
(see Theorem 7.3). Therefore, we have solved problem (9.29) —(9.30)
which did not fit the variational formulation of Section 9.5, with A = identity
(see Remark 9.7). Here, we have taken A = —A. U
Remark 9.11. If Q is sufficiently regular, v e V implies that Av e
eL2(Q) and we may take feL2(Q) in (9.41). D
10. Comments
We shall give few bibliographical references for equations of the second
order and refei the reader to the books of Bitzadze [4], Courant-Hil-
bert [1] and C. Miranda [1].
For equations of higher order than the second, boundary value
problems have only been studied recently (except for some particular cases
such as, for example, the iterated Laplace operator or equations in two
variables, for which the reader can consult E. E. Levi [1]), and it is
mainly following the works of Garding [1] and Vishik [1] that the
variational theory has been developed: various accounts of this theory
can be found in Agmon [7], Berezanski [4], Browder [3, 9], Kato [5],
Lions [2], Magenes-Stampacchia [1], Necas [2], Vishik-Ladyzenskaia [1].
We also note the work of Hestenes [1]. Here, we have only sketched the
outline of the theory.
Theorem 9.1 has been introduced and used by many authors: Lax-
Milgram [1], Vishik [1], . . .; for various extensions to "convex sets",
see Stampacchia [1] and then Lions-Stampacchia [1].
For the coerciveness problem pointed out in Section 9.6, we add the
works of Necas [2, 3], Schechter [1], Smith [1] to the references to
Agmon, Aronszajn and Smith made in the text.
The question of the regularity at the boundary of solutions of
"regular" variational problems (see Section 9.7) was resolved by Niren-
berg [2] (see also Guseva [1]). Also note the "compensation method"
of Aronszajn-Smith, an account of which is given in Lions [4], for
example. It concerns problems for which there is a certain regularity
of the data (coefficients of A, boundary conditions, open set Q).
If these conditions of regularity are not satisfied, the question of the
regularity of the solutions is much more difficult; for example, if the
coefficients of A belong to Z,00 (Q) the case of second order equations is
resolved in a satisfactory manner, thanks to the results of De Giorgi [1]
and Nash [1] and many other authors: H. O. Cordes, D. Gilbarg, O.
Ladyzenskaia, W. Littman, C. Miranda, C. B. Money, J. Moser, J. Servin,
G. Stampacchia, N. N. Uraltzeva, H. F. Weinberger . . .; we refer the
reader to Ladyzenskaia-Uraltzeva [1], Miranda [1], Morrey [2],
Stampacchia [2] and to the bibliographies of these works; for "mixed" and

10. Comments
217
"transmission" problems, see also further on. For the case of higher
order equations we note the recent results of De Giorgi [2], Money [5].
When the coefficients of A belong to L™{Q), the solutions can not be
generalized to spaces of functions with locally integrable first derivatives
without losing the fundamental properties; see Serrin [1].
Still in the setting of variational theory, we note that the idea
described in Section 9.8 was introduced in a general abstract manner
in Lions [2] and developed for many concrete problems by various
authors. For example, it is a well-known situation for the equations of
elasticity. It is also a natural situation for boundary value problems for
the hypoelliptic operators associated with the spaces H"'P (Q) and pointed
out in Section 9.8; see the works of H. Marcinkowska, V. P. Milkhailov,
P. P. Mosolov, S. N. Nikolski, M. Pagni, B. Pini cited in Pagni [1], and
also Pagni [2], Pini [12], Ramazanov [2].
For the problems of Section 9.8, see also Magenes-Stampacchia [1],
Pulvirenti [1, 2].
We call attention to Remark 9.7 and Section 9.9, which, we believe,
may be developed more systematically to obtain more precise
information on boundary value problems which may fit a variational theory.
The idea of replacing "A u = f" by (9.31), with A to be chosen, has
been used in many situations. For hyperbolic evolution equations this
idea plays an absolutely fundamental role; see the method called
"method a-b-c" of Friedrichs [3] and the obtainment of the energy
inequalities in Leray [1]. For elliptic equations, the notion of conditional
ellipticity (inspired by Leray [1]), which rests on a neighboring idea,
was introduced in Lions [8]; see also an analogous idea for ^-positive
definite operators: Martyniuk [1], Petryshin [1]. Along closely related
lines, see Chapter 4 of Berezanski [4] and the works [1, 2] of this
author. See also Dezin [1].
The example given in Section 9.9 reconsiders, in a clearer form, one
of the examples given in Lions [3].
We also note the following: if A (resp. A*) is an isomorphism of
D(A) (resp. D(A*)) onto H, then A is (by transposition) an isomorphism
of H onto D{A*)' and therefore (by interpolation) of [D(A),H]1/2
onto [D{A*),H][/2. In the case where [D[A),H]U2 = [D{A*),H]l/2
(for the study of this question see Kato [4, 5], Lions [17], Shimakura [1]),
setting V = [D (A), H]1/2, we see that A is an isomorphism of V onto V
and therefore A is associated to the sesquilinear form
a(u,v) = (A u,v) = scalar product of A u and v in the
anti-duality between V and V.
We also point out the use made by Necas [2, 4] of the equality of
Rellich for elliptic boundary value problems.

218 10. Comments
For relations between variational theory and non-variational theory,
see also Agmon [4], Shimakura [2].
We come now to the non-variational theory developed in Sections 1 — 7
of this chapter for regular elliptic problems.
The conditions of proper ellipticity of A and of system {Bj} covering A
(Definitions 1.2 and 1.5) were introduced by Lopatinski [1] and
Shapiro [1] and by Agmon-Douglis-Nirenberg [1]; the terminology "proper
ellipticity" and "covers", which we have used, is that of Schechter [2].
Definition 1.4 of normal system and Theorem 2.1 on Green's formula
are due to Aronszajn-Milgram [1]; see also Schechter [2] for the proof
as we have given it. The proof of Theorem 2.2, given in Section 4.3, is
due to Agmon.
For the existence of local solutions, the regularity at the interior
and the hypoellipticity of the elliptic operators (Section 3), starting
from the lemma of Cacciopoli [1]—Weyl [1], see John [1], Friedrichs [1],
Petrowski [1], Schwartz [7] (whose idea of the proof we have followed)
etc.
The characterization of hypoelliptic operators with constant
coefficients was given by Hormander [1]; for sufficient conditions for the
hypoellipticity of operators with variable coefficients see Friberg [1],
Hormander [2, 6], Malgrange [1], Mizohata [2], Treves [3]; a very general
sufficient condition for second order operators was obtained by
Hormander [11].
The direct a priori estimates (of the type of formula (4.39)) were
obtained by numerous authors: Agmon-Douglis-Nirenberg [1], Browder
[1, 2], Hormander [3], Koselev [1], Peetre [1], Schechter [1], Slobo-
detski [2, 3]; the use of the Fourier transform to obtain them, as we
did in Section 4, is classical by now.
To prove the existence of the index of the operator 0* in Sections 4
and 5, we have used the method of Peetre [2] which depends on the
obtainment of the "dual" estimates (formulas (4.40) and (5.4)) (see
also the account of Grisvard [3]). For Section 5.3, see Schechter [11].
We have pointed out the advantages of this method in Section 8.4. But
here, we have to recall the other methods which may be used to obtain
the final results of Section 5, by also generalizing them to Sobolev
spaces of Lp type.
The "direct" a priori estimates may be obtained in Lp by the method
of Poisson kernels which uses the theory of singular integrals of Calderon-
Zygmund [1] (see Agmon-Douglis-Nirenberg [1], Agmon [3], Browder
[1,2], Nirenberg [3]) and also by a method of Peetre [13] and Arkeryd [1].
We also call attention, for equations of variational type, to the estimates
in the spaces <£2'A (Q) of Morrey type (see Stampacchia [3] and the
bibliography of this work, for theses spaces) given by Campanato [3, 4, 6]

10. Comments
219
and from which the estimates in the spaces LP(Q) can be deduced (see
Campanato-Stampacchia [1], Giusti [1]), as well as certain results for
problems with discontinuous coefficients (see Campanato [6], Kadlec-
Necas [1]), which connects with the work of Morrey [1,4] and Cordes [1].
The existence of the index of 0> may also be shown by the
construction of a parametrix: see, for example, Chapter X of Hormander [6]
and Agranovich-Vishik [1]. For the case of problems with normal
boundary conditions, the existence of the index of 0> may be proved by
using the formal adjoint problem {^4*, C) directly and certain results
of variational theory (see Berezanski [4], Schechter [2, 3]).
In the case of the plane, we need to recall the methods which,
taking their inspiration from the classical theory for second order equations,
have studied the boundary value problems by using a global
representation of the solutions by "double or single layer potentials" and the
theory of singular integral equations in one variable (see Agmon [1],
Fichera [2], Muskhelisvili [1], Vekua [1], Volpet [1], etc.).
Finally, the theory of pseudo-differential operators (which generalize
both the differential operators and the singular integral operators),
developed and applied by Agranovich [1], Agranovich-Dynin [1],
Boutet de Monvel [1, 2], Calderon [4, 5], Calderon-Zygmund [2],
Dynin [1, 2], Grusin-Vainberg [1], Hormander [7, 8, 9], Kohn-Niren-
berg [1, 2], Mihlin [1], Shamir [4, 5], Seeley [1, 4, 5], Unterberger-
Bokobza [1], Vishik [5,) Vishik-Eskin [1—4], etc., also contains the
"regular" elliptic boundary value problems; then, the results of
Section 5 become particular cases of the results on pseudo-differential
operators. Even the boundary value problems for elliptic equations which
are not "regular" enter this theory; for example the general problem
of the "oblique derivative" and the "mixed" problem (see also below).
But we have not touched upon the questions pertaining to pseudo-
differential operators. This would have implied some very considerable
supplementary developments; and a great number of questions still
seem to be open in this direction (see the problems for this chapter).
Sections 6 and 7 of this chapter systematically develops and
completes (we have eliminated the exceptional values of the parameter 5
and introduced the spaces 5s(Q)) the work of Lions-Magenes [1].
The method of transposition for boundary value problems was
introduced by Sobolev-Vishik [1] and Fichera [1] and also used in
Lions [30], Magenes-Stampacchia [1], Berezanski [5]; it was
systematically applied to elliptic boundary value problems, furthermore
using interpolation and Sobolev spaces of Lp type (1 < p < oo), by
Lions-Magenes [1], then by Baiocchi [1], Barkoski-Roitberg [1], Beren-
zanksi [3, 4], Berenzanski-Krein-Roitberg [1], Berenzanski-Roitberg [1],
Geymonat [1], Roitberg [1, 2], Schechter [4, 8, 9].

220
10. Comments
In order to separate the equation A u = / from the boundary
conditions BjU = gj in the functional equation (6.6), we have been led
(see Lions-Magenes [1]) to the density and trace theorems (Theorems 6.4
and 6.5).
This enables us to give an interpretation "of the classical type"
for the resolved problems.
In Roitberg [3], the author obtains isomorphism results in spaces
obtained by completion of 2{D) for suitable norms. In orde* to
"separate" the various data (in Q and on U), he uses the trace theorems
(Lions-Magenes [1] and Section 6 of this chapter), which enables him
to obtain our results again (up to technical variants) and other results
of a more abstract nature.
In Section 7, the use of Theorem 14.3 of Chapter 1, due to Baiocchi [5],
enabled us to considerably simplify the account we had given in previous
works. Let us also mention that we have arrived at Proposition 7.1 and
Theorem 7.1 after discussions with G. Geymonat. For other results of
the type of Remark 7.3, see Lions-Magenes [3].
The property of continuity of traces on surfaces neighboring r
(Section 8.1) draws upon the formulation of boundary value problems
for elliptic equations of the second order given by Cimmino [1, 2], and
for polyharmonic equations by Sobolev [1] (see also Magenes [1],
Pini [4]).
The realization of A in L2 (D) (Section 8.4) is due to Browder [1, 5];
our proof follows Grisvard [3]. Concerning the vanishing of the index
(Fredholm operators, see result (8.14)) see Agmon-Douglis-Niren-
berg [1], Agranovich [1], Browder [6], Geymonat-Grisvard [1], Kaniel-
Schechter [1], etc.
For uniqueness theorems different from those of Section 8.3, see
Vishik [1] and Pini [3] (case of sufficiently "small" domains) and Agmon-
Douglis-Nirenberg [1] (case of "weakly positive semi-definite"
operators).
The technique of trace theorems given in this chapter is equally
useful for unilateral problems of which we present a simple example:
in'an open set Q with regular boundary r, there exists one and only
one function u e H1 (Q) such that
-Au + u = /, feL2{Q)
u^O on r (in the sense: y0 u ^ 0 in Hll2{D)
du
-— ^0 on/1 (in the sense: Yl u ^ 0 in H~ll2(r))
ov
du
u —— =0 on r (the multiplication u, v -> u • v is
ov

10. Comments
221
is a continuous mapping of H1/2(r) x H~1/2(r) -> $)' (T), for example,
du \
so that u has meaning)', for these problems, see Lions-Stampac-
dv J
chia [1] and the bibliography of this work.
Furthermore, the techniques of this chapter enable us to prove
uniqueness theorems of the following type: let R+ be the half-space
{xn > 0} and u an element of H~k(Rn+), k an arbitrary positive real
number, solution of
—Au + u = 0
which satisfies
(u {%', xn) - u {%', 0)) -+0 in S' (Rn-l) as xn -► 0.
Then u = 0 (indeed, it follows that —— = 0 in ^^-^(R"-1),
I dxn {
whence the result . This type of result extends to elliptic operators
with variable coefficients of arbitrary order and in the spaces Lp by using
Lions-Magenes [1] (V). For results of this type via different methods,
see Butzer [1].
Finally, we point out some other interesting questions pertaining
to elliptic boundary value problems which we have not studied in this
book:
1) The "topological" questions pertaining to the theory of boundary
value problems (study of invariants by "homotopy", computation of
the index, relations with certain classical problems of algebraic
topology) which have been posed and developed in the last few years and
for which we refer the reader to Agranovich [1], Atiyah [1], Atiyah-
Bott [1], Atiyah-Singer [1], Calderon [8], Gelfand [1, 2], Gohberg-
Krein [1], Palais [1], Seeley [1, 4], Volpert [2].
2) The "mixed" problems (pointed out in Section 9.5, example 3,
in the setting of variational theory): see Peetre [3], Schechter [5],
Shamir [1, 3-7], Vishik [5], Vishik-Eskin [1-3], etc.
3) The "transmission" problems (pointed out in Section 9.5,
example4, in the setting of variational theory): see Campanato [1],
Lions [31], Roitberg-Sheftel [2], Schechter [6], Sheftel [1—3], Stampac-
chia [4], Troisi [1—3], etc.
4) The general oblique derivate problem in any dimension: see
Bitzadze [2, 3], Egorov-Kondrat'ev [1, 2], Hormander [8], Vishik-
Eskin [5], etc.
5) The boundary value problems in domains with "angular points";
see Hanna-Smith [1], Kondratiev [2], Necas [2], Volkov [1] and also
10) and 11) below.

222
10. Comments
6) The axiomatic theory of the potential and the theory of capacity
according to Brelot, Choquet, Deny; see the seminar notes of Brelot-
Choquet-Deny [1] and the "connection" made between this theory
and the boundary value problems for elliptic equations of the second
order with measurable coefficients (R. M. Herve [1], Littman-Stampac-
chia-Weinberger [1], Stampacchia [2]).
7) Generalizations of the theory of Sections 3 — 8 to Lp (Q) type
spaces, p 4= 2 (Sobolev spaces Ws,p (Q), Besov spaces BSt p (Q), Lebesgue
spaces HS,P(Q)] we follow the notations of Magenes [3], for example);
see Lions-Magenes [1] and the accounts of Magenes [2, 3] and Geymonat-
Grisvard [1].
8) Spectral theory (study of the spectrum of problems, eigenfunction
expansions, asymptotic distribution of eigenvalues, . . .): we refer the
reader to the books of Agmon [7] and Berezanski [4] and to the works of
Agmon [6, 8], Agmon-Kannai [1], Aronszajn [5], Browder [5, 8], Fichera
[4], Garding [2], Geymonat-Grisvard [3], Mizohata-Arima [1], Peetre [11],
Plejel [1], Seeley [6], Weinstein [1], etc.
9) Fundamental solutions, Green's functions; see Berezanski [4],
John [1], Hormander [6], Miranda [1], Courrege [1], Treves [5] ...
In this regard, we call attention to the study of Green's function of
fractional powers of elliptic operators: see Kotake-Narasimhan [1],
Sobolevski [2]. Moreover, this question is tied to interpolation spaces
(see Kato [3 — 5], Lions [17], Fujiwara [1]).
10) "Variational" equations of the second order
i.e. of the type £ Dp(apqDqu)\
with discontinuous coefficients apq: problem of the regularity of the
solution (which we have already mentioned in these "Comments"
and in Section 9.7), maximum principle, Harnack's inequality, "regular"
and "non-regular" points of the boundary, "removable" singularities
of the solution, "isolated" singularities of the solution . . . (see the
authors and the bibliography already cited in connection with regularity
problems).
11) "Non-variational" equations of the second order
i.e. of the type £ apDp\
with discontinuous coefficients ap, for which analogous problems to
those of 10) exist: see Alexandrov [1,2], Bers-Nirenberg [1], Miranda [3],
Nirenberg [1], Pucci [1], . . ., [4], Finn and Serrin [1], Gilbarg and
Serrin [1], Talenti [1—3], Cordes [1], Stampacchia [5] ...
12) Boundary value problems in weighted Sobolev spaces and for
equations which "degenerate" on the boundary: see Baouendi [1],

10. Comments
223
Geymonat-Grisvard [2], Kudryavcev [1], Morel [1], Murthy-Stampac-
chia [1], Necas [2], Oleinik [5], Vishik [3], Vishik-Eskin [2], and the
results of Baouendi-Goulaouic [2], elucidating, as a consequence, the
topological structure of Q){D).
13) "Non-local" boundary value problems; see Bade-Freeman [1],
Beals [1], Browder [7], Fishel [1], Freeman [2], Grubb [1], Peetre [1],
Schechter [10], Vishik [3]; see also the Appendix to Volume 2 of this
book and the works on the infinitesimal generators of Markov
semigroups (see Dynkin [1] and the bibliography of this work). Also, the
integro-differential operators of the second order, which come up in
"non-local" problems, appear as infinitesimal generators of Feller
semigroups on a variety with boundary; see Bony-Courrege-Priouret [1]
and the bibliography to this note.
14) Boundary value problems in unbounded open sets: see Baioc-
chi [1], Barros-Neto [1], Freeman [1], Kudrjavcev [2], Lax [2],
Lions-Magenes [1] (I), Miranda [4], Peetre [1, 2], etc.: in this regard,
particularly Beppo-Levi type spaces (see Deny-Lions [1]) and the
completion of @{Q), in these spaces (see Hormander-Lions [1]) are
used.
15) Singular perturbations: see Friedrichs [2], Huet [1] ... [5],
Oleinik [6], Peetre [4], Vishik-Ljusternik [1], etc.
16) Study of boundary value problems in the spaces Ck,0l(D) of
"holderian" functions (Schauder type estimates, . . .); see Agmon [9],
Agmon-Douglis-Nirenberg [1], Miranda [2], etc.
17) Uniqueness and unique extension problems: see Aronszajn [6],
Cordes [2], Heinz [1], Landis [1], Mtiller [1], Pederson [1]; for equations
of general type, see Calderon [7], Hormander [6].
18) Generalizations of the preceding questions to systems of elliptic
operators (containing, in particular, the classical systems of elasticity
and of Stokes).
We recall the main definitions pertaining to elliptic systems.
Consider the matrix operator given by
A = A{x;D) = \\ltj(x'9D)\\t i,j = 1 »,
where the /l7's are linear differential operators with coefficients defined
in D.
A is said to be elliptic in Q, according to Douglis-Nirenberg [1],
if there exist integers st, tt, i = 1, . . ., m, such that the order of lu
is st + tj (where we take lu = 0, if st + tj < 0) and if, denoting by
l%{x}D) the principal (or characteristic) part of lu and by A°(x,D)
the matrix || 1% (x, D) ||, we have, for all x e D and all f e R" and 4= 0,
L°(xJ) =detM°(*ff)|| 4=0.

224
10. Comments
The system of operators A is said to be properly elliptic if, in addition,
m
£ (s, + tt) = lr,
1 = 1
integer r > 0, and if, for all # e D and every couple of linearly independent
vectors £, £' in R", the polynomial L° (#, £ + r £') in r has r roots with
positive imaginary parts: rf (#, £, £'), . . ., rr+ (#, £, £').
If w ^ 3, every elliptic system is properly elliptic. We could also
consider more particular classes of elliptic systems (Petrowski systems,
strongly elliptic systems . . ., see, for example, Volevich [2]). We just
recall the definition of strongly elliptic systems: A is said to be strongly
elliptic in D if tt = st > 0 and if for every x e D and for every complex
vector A = (Ax, . . ., Am) and every £ e R", with A =t= 0 and £ =(= 0, we
have m m
Re I (-D**i2r(^.« Af^ ^ * Z |f|a'*|Af|2.
with k a positive constant.
We also recall the definition of systems of boundary operators
covering A (generalization of Definition 1.5 of this chapter). Given a properly
elliptic matrix operator A, we denote by «$/(#,£) the adjoint of the
matrix A°{x, £) (i.e. such that A°(x, £) s/{x, £) = L°{x, £) I, I =
identity matrix). We consider a matrix of linear differential operators with
coefficients defined in J1:
B=B(x,D) = \\BqJ(x,D)\\9 q=\,...,r, j = 1 i».
We say that B covers A if there exist integers aq> q — 1, . . ., r,
such that BqJ is of order aq + tj (if aq + tj < 0, we assume that Bqj = 0)
and if, denoting by B%J (x, D) the principal part of BqJ (x, D) and by
B°(x,D) the matrix ||B°qJ{x, D) ||, for every ^ef, every £eR"
tangent to r at x, every £r e R" normal to r at x, the rows of the matrix
B°(X,£ + T?).*/(X,£ + T?)
(the elements of which are polynomials in r) are linearly independent
modulo the polynomial
n(T-T»+(*,f.f)).
k=l
The Dirichlet system covers every strongly elliptic system (see
Agmon-Douglis-Nirenberg [1]) but does not cover every elliptic system
(see Bitzadze [1]).
There is not yet a general definition of normal boundary operator
systems; the problem is tied to the validity of a Green's formula, analogous
to formula (2.3), for systems. We refer the reader to Geymonat [3],
Lipko-Eidelman [1], Roitberg-Shefter [1] for results on this subject.

11. Problems
225
Many results for elliptic equations have been generalized to elliptic
systems: in particular, the a priori estimates and the existence of the
index, in L2 as well as in Lp, for boundary value problems {A, B),
with A properly elliptic in the sense of Douglis-Nirenberg and with B
covering A; but many other problems still remain unsolved on this
subject (see, for example, problem 11.1).
Consult Agmon-Douglis-Nirenberg [1], Agranovich-Dynin [1], Agra-
novich-Dynin-Volevic [1], Avantaggiati [1, 2], Campanato [1], Can-
fora [1], Cattabriga [1], Douglis-Nirenberg [1], de Figuereido [1], Gey-
monat [2, 3], Gobert [1, 2], Hormander [6], Lawruk [1], Morrey [1, 2],
Necas [1, 2], Pini [2], Roitberg-Sheftel' [1], Sheftel' [1], Solonnikov [1],
Vishik [1, 5], Vishik-Eskin [1, 3], Volevic [2], Volpert [1], etc. Pseudo-
differential systems are studied in Vishik-Eskin [5].
19) Elliptic systems of the first order which generalize the Cauchy-
Riemann system, the theory of pseudo-analytic functions and quasi-
conformal representation: see Bers [1], Hormander [10], Lavrentiev [1],
Miranda [5], Morrey [3], Stampacchia [6], Vekua [2].
20) Problems pertaining to the approximation of the solution by
finite difference methods: there exists a vast literature on this subject,
especially for homogeneous variational problems, but the study of
the general case (even from just the point of view of convergence) and
the estimate of the error still cause numerous problems. Particular results
may be found in Bramble [1], Jamet [1], Lions [29].
Finally, questions concerning analytic regularity, or regularity in
Gevrey classes, of solutions and the study of boundary value problems
in classes of distributions or ultradistributions will be investigated in
Volume 3 of this book.
11. Problems
11.1 Extension of this chapter's results to elliptic systems.
For problems {A, B), with A properly elliptic in the sense of Douglis-
Nirenberg and B covering A, the a priori bounds are known (see 18)
in the Comments); the essential difficulty resides in Green's formula,
for the results of Geymonat [3], Lipko-Eidelman [1] and Roitberg-
Sheftel' [1] are still incomplete.
11.2 Is it possible to obtain the results of Section 7 directly,
without interpolation! Among other things, this would no doubt allow
a weakening of the regularity hypotheses on the coefficients (see
Campanato [2]).
11.3 In the extension of the theory of Sections 6 and 7 to spaces
constructed on LP(D), p =t= 1, 2, oo (see Lions-Magenes [1], (III-VI)),

226
11. Problems
is it possible to avoid the "exceptional values" of 5 (as was done here
for the case p = 2) ?
Similarly, it would not be without interest to extend the theory of
Ss(i2)-spaces to "analogous" SS,P(Q)-spaces constructed on LP(Q),
p =t= 1, 2, oo (see Problem 18.4, Chapter 1).
11.4 Regularity results on Lipa-spaces are known [see 16) in the
Comments]. "Abstractly", the methods presented here fit. But the
interpretation of the abstract results leads to new interpolation problems
which seem delicate.
11.5 Development of the idea mentioned in Section 9.9: which
boundary value problems fit the variational setting, after introduction
of a "suitable" operator A (see Remark 9.7)?
11.6 Systematic study of boundary value problems for quasi-
elliptic operators for which we already know the inequalities "in the
interior", in L2 (see Friberg [1], Hormander [6], Pini [8], Volevich [1]),
in Lp (see Kree [4], Giusti [1]), in J^2, A (see Giusti [1]) and the inequalities
in the neighborhood of the subsets of the boundary which are not
"singular" with respect to the operator (see Cavallucci [1], Matsu-
zawa [1, 2]; see also, for parabolic operators, Chapter 4, Volume 2 of
this book); see also Pagni [1, 3], Pini [9, 11], Ramazanov [2].
11.7 Use of pseudo-differential operators for regular,
non-homogeneous elliptic boundary value problems for all values of 5 (see
Hormander [8], Vishik-Eskin [4, 5]). Maybe transposition can be avoided,
but, in order to interpret the problems and for the choice of /, suitable
trace theorems, the study of which does not seem to have been initiated
yet, will be required.
11.8 Study of non-homogeneous boundary value problems for
pseudo-differential elliptic operators.
V
11.9 We may consider the open sets with boundary 71 = [j Tit
where the rt's are varieties of dimension less than n — 1 (see Stermin [1]).
11.10 Non-homogeneous problems of "transmission" type (see
Section 9.5, example 4 and 3) in the Comments).
11.11 Questions pertaining to approximation by finite difference
methods (see 20) in the Comments): for the non-homogeneous problems
considered in this chapter, the question is open in general. See Lions [29].

Chapter 3
Variational Evolution Equations
Sections 3, 5.2, 5.3 and Remark 9.5 rely on Chapter 1. For the rest
of this chapter, the knowledge of Chapters 1 and 2 is not required.
Section 7 may be skipped on first reading. Sections 8— 10, with the
exception of 8.3, may be read independently of the rest of the chapter.
1. An Isomorphism Theorem
1.1 Notation
Let V and Jf be two Hilbert spaces, with Y cz Jf and Y dense in Jf;
|| || y, || || *> denote the norms in y and j?\ in order to simplify the
writing, the scalar product in Jf is denoted by ( , ). We identify 3f
with its antidual; then, if y denotes the antidual of y we have
fc/cf. □
Remark 1.1. Except for the notation, we have already seen examples
of this situation in Section 9 of Chapter 2. But in the present chapter,
the spaces y and Jf will "contain the time" — which was not the case
for the stationary problems of Chapter 2. □
If / e y and i/ef, their scalar product (antilinear in v) is denoted
by (/, v)\ it coincides with the scalar product in Jf when / e 34?. □
'As we have already agreed upon in the preceding chapters, (•, •)
and sometimes [•, •] denote sesquilinear scalar products (linear in /,
antilinear in v) and <•, •> denotes bilinear scalar products. □
The semi-group G(s)
We introduce a semi-group
(l.i) s-+g{s) of r+ -,^(y;y),
continuous and bounded in y: V/ e y, 5 -► G(s) f is a continuous
function of 5 ^ 0 -* y; we have
G[s)G{t)f = G{s + t)f, Vs,^0
G(0)/ = / and || G (s)|| jg>(y/.y/) ^ constant.
(1-2)

228 1. An Isomorphism Theorem
Remark 1.2. Consult Hille-Phillips [1] or Yosida [2] for the theory
of semi-groups. We shall use only the simplest results of the theory
(the same applies to Volumes 2 and 3). □
We make the hypothesis:
1 G(s) forms a bounded, continuous semi-group in Y°\
I therefore Vs ^ 0, G{s) E^{i^\ "T) and Vv e *T,
I s -+G(s) v is a continuous function of 5 ^ 0 -* "P" with
I properties analogous to (1.2).
Remark 1.3. From (1.1), (1.3) and the fact that [T, ^']1/2 = #
(Chapter 1, Section 2.4), we obtain that G(s) is a bounded, continuous
semi-group in J^ (Section 10.4 of Chapter 1, is also necessary to obtain
this result). □
Furthermore, we shall assume that G(s) is a contraction semi-group
in Jf:
(1-4) \\G(s)\\*i*;*)£ 1-
We summarize the hypotheses by:
G (s) is a bounded, continuous semi-group in i^'
and 'V (therefore in 3t?) and \\G(s) \<ew;#) ^ 1, s^O. □
We denote by —A the infinitesimal generator of G (s) and by D [A; y)t
D(A,3f), D{A\ir) its domain in the spaces iT\ Jf\ Y.
For example, the space D{A\ *¥*') is provided with the norm
(IM£. + M»|£.)1/a.
which makes it a Hilbert space (since A is closed); similarly for D {A; Jt?)
and D(A\-r).
Therefore, we have
(1.6) *T n D(A\ f')cf c^cf
and
Lemma 1.1. In (1.6), each space is dense in the following one.
Proof. We only need to verify that V n D{A\ ir') is dense in ir\
but
fnD(/l;f') DD(yl;f)
and
D{A\ir) is dense in tT. D
Since G (s) is a contraction semi-group in Jf, we have (consider the
derivative at the origin of the function 5 -► \\G(s) v\\]r):
(1.7) R[Av,v) ^0 VveD{A',JP).
(1.5)

1.1 Notation 229
We also have
Lemma 1.2. Under hypothesis (1.5):
(1.8) Re{Av,v) ^0 ¥i;efnD(yl;f').
Remark 1.4. If veD(A\ IT'), then AvelT' and, in (1.8), {A v, v)
denotes the scalar product between 'V and "T. □
Proof of Lemma 1.2. 1) Let <p be a continuous (scalar) function with
compact support in £ ^ 0. We denote by G(q>) the operator in y'
(resp. iT, resp. Jf), given for / e ir' (resp. ^, resp. Jf) by
00
G(<p) f = f G(s) f 'q>(s)ds,
0
the integral being taken in 'V (resp. "Tt resp. Jf). If, furthermore, cp is
once (resp. infinitely) differentiable, G(cp) f is inD(A) (resp. Z) (/I00)) and
^G(y)/ = G(?>')/.
2) Let @„ be a regularizing sequence of functions of 9) (R) with support
in t ^ 0 (see Schwartz [1]).
We easily verify that
Gy^-^v in iT, Vve'f*,
AG{q„)v-+Av in «r>', ^veD{A'tr')
and therefore
{Av,v) = \im(AG(Qn)v,G(Qn)v), Vv er c\D(A\r').
n-»oo
Now, G{Qn)veD{A\ir)^D{A\^>) and, according to (1.7), we
therefore have
Re(/l G[qh) v, G{gn) v) ^ 0; whence (1.8). D
The operator M
We introduce an operator M, satisfying:
Me^(f;f),
Re(Mv,v) ^ * ||i;||*f oc > 0, V^ e tT.
77&m M zs aw isomorphism of "T onto V (indeed, it is easily verified
that the image of "T under M is dense and closed in irr). D
The Problem
We consider the (operator) equation:
(1.10) Au + Mu = f,
where f e'V is given, arid we seek a solution u e V n D [A; y).
(1-9) , „ „ .

230
1. An Isomorphism Theorem
Note that
A + M e&(r nD[A',1rt)','rt).
1.2 Isomorphism Theorem
Theorem 1.1. Under hypotheses (1.5) and (1.9), the operator A + M
is an isomorphism of "T n D{A\ ir') onto V.
Remark 1.5. Examples are given in Sections 4 and 5. In the
applications, A is a differential operator in t (time-variable) and M is an
"elliptic" operator in the space variables. □
Remark 1.6. Abstract operator equations of the form (1.10), but in
a more general setting and with techniques different from the one we
shall use, are studied by P. Grisvard [5 — 7, 9] and, with still different
techniques, by Da Prato [4, 5]. □
The proof of Theorem 1.1 is given in Section 1.4. First, we make
some remarks concerning the adjoint /I* of A.
1.3 The Adjoint A*
Let G*(s) be the adjoint of G(s) in the sense:
(1.11) (G*(s)f,g) = (f,G(s)g), Vfer'. ger.
Then, since G{s) e^C(^'; V) n J2?(f; -fT) n „S?(jf; jf), G*{s) has
the same properties and we have
iiG*(s)b(jr;jr)<n.
Let —A* be the infinitesimal generator of G*(s) in "T' (and in Jf
and iT) and let D(A*\ir')9 D(A*;J*r), D{A*)ir) be its domain in
*r9 #e, r.
The operator A* is the adjoint — in the sense of unbounded operators
- in iT' (resp. ^, resp. iT) of A in iT (resp. 2tf 9 resp. V) and vice-
versa (see Hille-Phillips [1]). Therefore we have
Lemma 1.3. The necessary and sufficient condition for given u ei^ to
he in D(/I*; i^) is that v -+ (Av,u) be continuous on D(A; i^') in the
topology induced by if*.
Analogous results hold on interchanging V and V and also A and A*.
1.4 Proof of Theorem 1.1
It is sufficient to show that equation (1.10) has a unique solution.
1) Uniqueness
Assume / = 0 in (1.10). Then
0 =■ Re(Au + Mu,u) ^ Re(Mu,u) ^ «||«||J-,
by Lemma 1.2 and (1.9). Therefore u = 0.

2.1 Generalities
231
2) Existence
For wef and v e V n D (A*; f'), we set:
(1.12) E(u,v) = (u,A*v) + (Mu,v).
According to Lemma 1.2 (for A*, which is allowed) we have
Re{v,A*v) ^0, VvefnD(/l*;f')
and therefore
ReE(v,v) ^<*\\v\\i.
By a variant of the projection lemma (see Lions [13], Chapter 3)
(see also a different proof in Section 7 below) there exists u ei^, with
(1.13) E(u,v) = (/,»),. VvernD(A*\r').
We show that this implies that u eD(A] V); indeed, we may
write (1.13) in the form
[u,A*v) = (/ - M u,v)
and therefore v -> (u, A* v) is continuous on "T n Z)(/l*; y*') in the
topology induced by y, therefore certainly continuous on D(A*; °T)
in the topology induced by "T and therefore, according to Lemma 1.3,
usD(A\ir') and
(Au,v) = (f - Mu,v), Vve^ nfl(^;f').
Since y n Z)(/l*; y') is dense in "T, we obtain that u satisfies (1.10);
hence, the theorem is proved. □
Remark 1.7. At this stage, the reader can consult the examples at
the beginning of Section 4. D
2. Transposition
2.1 Generalities
We have already used transposition for elliptic problems in Chapter 2,
Section 6. Here, we reconsider the general idea in a heuristic way.
Let E, F be two topological vector spaces imbedded in the same
topological vector space s/ and let U be an operator which is known
to be an isomorphism of E onto F.
Let U* be the "adjoint" of U (in j/) which is known to be an
isomorphism of Ex onto F1 (Elt F1 are spaces imbedded in s/). Then
by transposition, U** is an isomorphism of F[ onto E[.
But under reasonable hypotheses, U and U** coincide on a "dense"
subspace; we may extend U from F[ to E[ and this extension coincides
with [/**.

232 2. Transposition
Thus U is an isomorphism of E -+ F and of F[ -+ E[.
Then we can interpolate.
We shall apply this (in a more rigorous fashion!!) to the situation
of Section 1.
2.2 Adjoint Isomorphism Theorem
Since M e&(ir\ir'), its adjoint M* e .2? (f; f') and
(2.1) Re(M*v,v) ^a|M£, V^e^.
Therefore, according to Theorem 1.1, we have:
(2.2) A* + M* is an isomorphism of "K n D (/I*; i^') onto y.
2.3 Transposition
With the symbol ' still denoting passage to the antidual, we deduce
from (2.2) that
(2.3) (/I* + M*)* is an isomorphism of tT onto (tT n D(A*; tT))'. D
Since f nD(/l*;f') c f c «f c f, each space being dense
in the following one, we have:
(2.4) r n D(A*\ iT') c f c jf c tT' c(tT n £>(/l*; 1T'))'.
If / 6 0^ n Z) (4*; f'))' and i; e ^ n Z) (/I*; iT'), their scalar
product is denoted by (/, v), since it coincides with the scalar product of
the antiduality between V and iT' if / e ir'. D
We may define A by extension on "T (M is already given on y),
for
therefore
(/I*)* eif(^; (^ nD(4*; iT'))')
and obviously (/I*)* v = A v ii v e i^ n D(A; V). Therefore
(2.5) /I e &{T\ (T c\D(A*; r'))')
and we immediately verify that
(/I* + M*)* = /I + M.
Therefore, we have
Theorem 2.1. Under hypotheses (1.5) and (1.9), the operator A + M
is an isomorphism of V onto (y n Z)(/l*; y'))'.
We shall now interpolate "between'' Theorems 1.1 and 2.1.
Remark 2.1. As before, we interpolate only by the method of traces
of Chapter 1.

3.2 Characterization of the Interpolation Spaces 233
3. Interpolation
3.1 General Application
From Theorems 1.1. and 2.1, we deduce
Theorem 3.1. Under hypotheses (1.5) and (1.9), the operator A + M
is an isomorphism of
\TnD(A\ r'), r]e -+ \fl\ {T n D {A*; r'))*],
for every 6 with 0 ^ 6 ^ 1.
There remains to characterize the spaces which appear in this
theorem.
3.2 Characterization of the Interpolation Spaces
First, we have
Proposition 3.1. Under hypotheses (1.5) and (1.9), we have
(3.1) \T n D(A; r'),r]9 = <T n [Z)(/l; ^)^]a.
Proof. We denote by £ and F the spaces of the first and second
member of (3.1) respectively.
Since obviously E a F, we need only show that F cz E.
Now
(3.2) A + Me £{F\ \T' ,{T nD[A*; *"))'],) •
Indeed
Ae&(D{A\ r')\r') n £{T\ {T nD{A*', r'))')t
therefore, by interpolation1:
A e &([D [A; r1), r]B; \f9 (f n D {A*; *"'))']•)
and therefore a fortiori:
/I eif(F; [*", {T nD(A*; ^'))']e)-
AndM e if (TT; f')f therefore Me if (F; pr\ (f nD(yl*; *"))']•).
whence (3.2).
Then, let feF\ according to Theorem 3.1, there exists a unique
w e E such that
(A + M)w = (A + M) /, therefore {A + M) (w - /) = 0.
Since w — fe'f* and according to Theorem 2.1, it follows that
w - / = 0.
Therefore f eE, i.e. F a E, whence the proposition. □
1 The definition of the space [X, Y]e is valid without the inclusion of X in Y
(or vice-versa).

234 4. Abstract Parabolic Equations, Initial Condition Problem (I)
Proposition 3.2. Under hypotheses (.1.5) and (1.9), we have
(3.3) \T'9 {r nD(A*i r'))'\ = (r n [Z)(^*; f')f f]i-*)'.
Proof. According to the duality theorem (Chapter 1, Theorem 6.2),
we have:
\T\ (T n D(A*; *"))'], = [^ ^ n Z)(/l*; *")];.
Applying Proposition 3.1 (with /I* replacing /I) this is equivalent to
(rn[r,D(A*ir%y
and since
[r,D(A*;r')]9 = [D(A*:rf),r]i-e.
(3.3) follows. □
With the help of Propositions 3.1 and 3.2, Theorem 3.1 may be
stated in the following form:
Theorem 3.1a. Under hypotheses (1.5) and (1.9), the operator A + M
is an isomorphism of &e = "T n[D(A; ir')i lT}e onto (#£""•)', where
0l = rn[D(A*-fr')tr]e
(for 0^8^1).
3.3 The Case "0 = 1/2"
We could ask whether it is possible to find a space W such that
A + M is an isomorphism of W onto its antidual W.
The following is a partial result:
Theorem 3.2. Assume that (1.5) and (1.9) hold. Also assume that
(3.4) [D{A;r')t<r]1/2 = [D^jf'),^.
TAew ^1 + M is an isomorphism of &1/2 onto (01/2)'.
Proof. Apply Theorem 3.1 a with 6 = \. But &IJ2 = 01/2,
according to (3.4), whence the result. □
4. Example: Abstract Parabolic Equations,
Initial Condition Problems (I)
4.1 Notation
We consider the setting of Chapter 2, Section 9.1.
Let V and H be two Hilbert spaces, V <= H, V dense in H\ let || j|
and | | denote the norms in V and H and [, ] the (sesquilinear) scalar
product in H. We identify H with its antidual; then
VcHcV.

4.2 The Operator M
235
If / e V and v e V, [f,v] denotes, in this chapter (there is a slight
difference with the notation of the other chapters), their scalar product
in the antiduality. D
Let t be the time variable. We shall assume that
te[0,T], T finite or +oo.
With the notation of Section 1, we take
(4.1) r = L2(0, T\ V), JP = L2(0, T; H),
where, if X is a Hilbert space, L2 (0, T\X) denotes the space of (classes of)
measurable functions / of (0, T) -* X for the Lebesgue measure dt,
and such that
(T \ 1/2
f 11/00 IIx di) < oo (see Chapter 1, Section 1).
Then we have
(4.2) r9 = L2(0,T;P).
4.2 The Operator M
For t e [0, T], let # (/; u, v) be a continuous sesquilinear form on V.
In order to slightly simplify the measurability considerations, we
assume V to be separable.
The conditions on the family a[t\u,v) are as follows:
f Vu,v eV, the function t -+ a{t\ u, v) is measurable
(4.3) J and
I \a(t',v,u)\ <; c \\u\\ \\v\\, c= constant, We [0,7],
{(uniform coerciveness on V) there exists a A such that
Rza{t\v}v) + AM2 ^*|M|2, *>0, VveF, V/e[0,T].
Since, for fixed t, the antilinear form v -> a(t;u,v) is continuous
on V, we have:
(4.5) a[t\ u, v) = [4 (t) u,v], A (t) u e 7',
which defines
(4.6) A{t)e&(V\V).
In this section, we want to solve the equations
du
A{t)u + = /, 0 < t < T
dt
u{0) =0,
under suitable hypotheses on u and /.

236 4. Abstract Parabolic Equations, Initial Condition Problems (I)
We may always make the change of functions
u = e~At w
and obtain the equivalent equations
dw
(A{t) + XI)w+ = (e~At /), 0 < t < T,
dt
w{0) = 0.
At the risk of changing A (t) to A (t) -f A /, we may always consider
the case where A (t) verifies (4.4) i with A = 0; therefore, in the sequel,
we assume that
(4.4) Rea{t;v,v) ^ oc \\v\\2, oc> 0, V^eF.
[We must be careful about the changing behavior at infinity if we
use the same classical device for 0 < t < oo.]
For vef, we define:
(4.7) M v = function "t -+ A [t) v(t)'\
or
(4.7 a) Mv{t) = A{t)v{t) a.e.
We have:
Lemma 4.1. Under hypotheses (4.3) —(4.4), the operator M defined
by (4.7) satisfies condition (1.9).
Proof. For u, v e Y* = L2 (0, T\ V), set
T
(4.8) Jt{utv) = [ a(t\u{t),v{tj)dt.
o
Thanks to (4.3), the function t -+ a(t; u(t), v(t)) is measurable and
bounded in modulus by c \\ u (t) \\ \\ v (t) \\, which is integrable, therefore
(4.8) has meaning and
\Ji{u,v)\ ^c\u\r\v\r.
The antilinear form
v -► Jt{u, v)
is continuous on V9 therefore in the form
J({u,v) = [NLu,v), JSLueY',
or
T
Jt{u,v) = [[Mu{t),v{t)]dt.
0

4.3 The Operator A 237
Therefore
&u(t) = M u(t) a.e.,
and
(4.9) Jt(uyv) = (Mu,v), Me&\f\ir')m
Finally,
T
Re{Mv,v) = f Rea(t',v(t),v(t)) dt ^ « ||v||*,
o
which proves the lemma. □
4.3 The Operator A
For the semi-group G(s) (see Section 1), we take the semi-group
of right translations (in t). More precisely, if fei^', we set
[0, if 0 < t < s
(4.10) G(s)f(t)=\ a.e. int.
[f(t-s), if s<t<T
We define, in this manner, a contraction semi-group in ir', V
and Jf.
Then, we have
du
(4.11) /U = = u'}
dt
(4.12) D{A-rf) = \u\ueir,> — ef'f «(0) = ol
[ ^/ J
where is taken in the sense of distributions taking their values
dt
in V'\ then u, after possibly a modification on a set of measure zero,
is a continuous function of t e [0, T] -► F', and therefore the condition
"«(0) = 0" is well-defined).
Of course, a description analogous to (4.12) holds for D(A\3f)
and Z)(^;tT). D
The adjoint semi-group G*(s) is defined by:
i) if T < oo,
f /(* + 5), if 0 < t < T - s
(4.13) G*{s)f[t) = a.e. in t\
\ 0, if T -s<t<T
ii) if T = + oo,
(4.14) G*(*)/W = /(* + «), *>0.

238 4. Abstract Parabolic Equations, Initial Condition Problems (I)
Then we have
du
(4.15) A*u =
dt
with
(4.16i) D(A*\«r') = b\ueir', — e'rt,u{T) = 0, if T < oo
( dt
(4.16ii) D(A*\«r') = b\uEr,t—er,l if J = + oo|.
( ^/ J
Analogous description for D [A*; Jf) and D(A*',ir). D
4.4 Application of the Isomorphism Theorems
The results of Sections 1 — 3 apply to the setting of this section.
The application of Theorem 1.1 yields:
Theorem 4.1. Assume that (4.3) — (4.4) hold and that A is given by
(4.11)-(4.12). Then, for given f e ^' = L2(0, T\ V), there exists a
unique u ei^ such that
(4.17) A{t)u + u' =f
(4.18) u{0) = 0.
Note that (4.17) implies that u' el,2(0, T\ V), so that
ueC°{[0tT]\H)
and u(0) is well-defined (Chapter 1, Theorem 3.1 and Proposition 2.1).
Proof. Indeed, Theorem 1.1 shows the existence and uniqueness
of u in
solution of (4.17). But then we have (4.18). Conversely, if u e "K and
verifies (4.17), then u' = / - A(t) ue^', which together with (4.18)
shows that ueDiA;^). □
Remark 4.1. In the examples (see Section 4.7), the A(t)'s will be
elliptic differential operators. Then the operators A (t) + djdt are
parabolic operators. □
Remark 4.2. In Volume 3, we shall consider problems of type (4.17)
to (4.18) with
A(t) = A, independent of t
and, furthermore, where —A is the infinitesimal generator of a
semigroup H (s) (which, in fact, we shall denote by G (s)). This semi-group
is of course essentially different from the semi-group for which —A
is the infinitesimal generator. □

4.5 Choice of L in (4.20)
239
Remark 4.3. Let u be given in H. Under the hypotheses of Theorem 4.1,
there exists a unique u satisfying (4.17) and
(4.19) u{0) =u0.
Indeed, theie exists (trace theorem, Chapter 1, Section 3) we'f,
with w' e V and w (0) = u0. Then u — w must verify
^ {t) (u - w) + (u - w)' = f - (A (t) w + w') = / e ^',
(u - w) (0) = 0
and we are led back to Theorem 4.1.
In the following chapters, we shall see how we may consider
conditions of type (4.19) with u0 taken from much more general classes.
The application of Theorem 2.1 yields:
Theorem 4.2. Assume that (4.3) —(4.4) hold and that A is given by
(4.11)-(4.12). Then, for L given in (i^ n D (A*; i^'))', there exists a
unique u e *¥* = L2 (0, T\ V), satisfying
(4.20) (u,A*{t)v - O = [L.v), VveirnD{A*',ir').
First of all, let us write out tT n D(A*\ irf):
\ rnD{A*;r') = \v \v e L2{0,T]V),
v' = — eL2{0,T] V), with v{T) =0 if T < oo ;
( dt J
so that the subspace ^(]0, T[; V) of C°°-functions with compact support
in 10, T[ and with values in V is not dense in y n D {A*; 1r') (its closure
is^nZ)(4*;ir')nZ)(4;iH).^
is not necessarily a distribution taking its values in V.
Here, we find again a problem of a type which we have already met
in Chapter 2 (and which we shall often meet again in the sequel): choose L
"in the best way" so as to "interpret (4.20) suitably".
4.5 Choice of L in (4.20)
Formally, at first, we take
T
(4.21) (L,v)=j V(t),v(t)] dt + [u0, »(0)],
0
where:
/ is a "function" given so that the integral "is well-defined",
u0 is given in a suitable space.

240 4. Abstract Parabolic Equations, Initial Condition Problems (I)
Then, still formally, u satisfies
IA It) u + u' = /
u(0) = u0.
Now, we have to make this rigorous. □
First of all, as v describes V n D (A*; 'V'), v (0) describes H and
therefore we must take
u0 e H.
The choice of / is more delicate.
We introduce the space S (compare with Chapter 2, Section 6):
let q be defined by
I tjt0} if 0^ t^t0>
(4.23) g(t) = t0 > 0, fixed.
[ 1, if t^.t0i
Then
(4.24) E= {v\veL2{0,T)V')=ir'>Qv'eir',v(T)=0(iiT<oo)}
(provided with the "usual'' norm
(ii^iir' + ii^'ii^)1/2.
which makes S a Hilbert space).
The space ^(]0, T[; V) is dense in S\ indeed let v be given in 3
and 6n be a sequence of functions belonging to C°[0, T],
en(t) = i if t*2S„. en(t)=o if tze. and \te'n(t)\^c,
with analogous conditions in the neighborhood of T if T < oo, where
£n is a decreasing sequence which tends towards zero.
Then, dnv-+v inS(immediate verification: @0^-»Oin L2(0,T;V'))',
we then regularize 6n v in t, whence the result. Therefore S' is a space
of distributions on ]0, T[ taking their values in V\ more precisely:
Proposition 4.1. Every j*eE' may be written — non-uniquely — as
(4.25)
d
fieL2(0,T;V).
Proof. The proposition is a consequence of the Hahn-Banach Theorem
and of the density of 3(]0, T[; V) in S. D
Since iT n D(A*] Y*') <= S, we have:
Proposition 4.2. In (4.20),* (L,v) may be chosen in the form
(4.26) (L,v) = (/*, v) + (/„, i;) + [uo,v(0)]t

4.6 Interpretation of the Problem 241
where:
/* is given in 3' (see Proposition 4.1),
/** is given in <T' = L2(0, T\ V),
u0 is given in H.
(In (4.26), [f*,v) — resp. [f**,v) — represents the scalar product
between 3' and 3 — resp. between V and ir). □
4.6 Interpretation of the Problem
Now, we need to make (4.22) more precise. If in (4.26) we take
v(t) =<p(t)k,
<pe9Q0,T\), keV,
then we deduce from (4.20) that
(4.27) A(t)u + u'=f in the sense of 9' (]0, T[; V), / = /* + /**.
Thus, if we define Y by:
(4.28) Y = {v\ver,v' eS' + f'}f
we see that weY (since u' = f — A(t) u and ,4 (/) « e y').
We have the following trace theorem in Y:
Theorem 4.3. The space £&([0, T]; F) zs rf^ws^ iw Y. 77^ mapping
u ->u(0) of £& ([0, T]m, V) -* 7 extends by continuity to a continuous linear
mapping, still denoted by u -+ u(0), of Y -+ H.
Proof. 1) Density. Let w->M(w) be a continuous antilinear form
on Y; it may be written
(4.29) MM = (g0/«) + (gltu')t g0erf, glernS,
where (g0> u) (resp. (glf «')) denotes the antiduality between irr and i^
(resp. between V c\S and ^' + £,/).
We assume that
M{q>) = 0 V<pe^([0,T];F)
and, from this, we shall deduce that M(u) = 0, Vw e Y.
Let g0, #! be the extensions of g0 and gx to Rt by 0 outside of ]0, T[.
Then V$e^(Rt;F), of restriction q> to ]0, T[, we have:
»o.*) + »i.*/)=M(y)=0
and therefore
(4.30) fo__?1 =0 in ^'(R,;n-

242 4. Abstract Parabolic Equations, Initial Condition Problems (I)
But g0 e!2(Rf; V), therefore necessarily:
(431) \gleL*(0,T;V), ^-eL2(0,T;F), gl(0)=0,
[ and gt(T) =0 if T < oo.
Thus, we can find gln e ^(]0, T[; F) with
(4.32) gl,-gl in L2(0,T;F), -^ - -^- in ^(0,7; F').
Therefore, m particular, gln -^ g1 in S n i^ and, if u e Y,
{gltW) = lim (gln,W) = lim - (g'ln>u)
and
-&..«)-*-&.«) (duality*",^).
So that
and
M(«) = (g0 - g'lf «) = 0, according to (4.30),
whence the density.
2) We shall now define u (0) by Green's formula (note that the idea
of this proof is entirely analogous to the proof of Theorem 6.5 of
Chapter 2).
According to Section 3.2 of Chapter 1, there exists a continuous linear
mapping:
h->vh
of H -► {v | v e r, v' e TT', v(T) = 0} such that
(4.33) 1^(0) -A.
For w given in Y, we set
(4.34) S(h) = -«vfc) - {u,v'h),
where («', vB) (resp. (u, v'k)) denotes the antiduality between S' + V
and S c\i^ (resp. ^ and ^')-
Formula (4.34) does not depend on the choice of the "right-inverse''
vh, as long as it satisfies (4.33). Indeed, if vh is a second "right-inverse",
w = vh — vh vanishes at 0 and T and then we see, as in 1), that
— («', w) — (u, w') = 0.
Furthermore, since the form h -► 3?(h) is continuous antilinear
on H, it may be written:
(4.35) &(h) = [ruth]} rueH,
which defines a continuous linear mapping u -+ r u of Y -+ H.

4.7 Examples 243
3) There remains to see that x u = u(0) if u e@([0,T]; V), which
follows immediately by integration by parts on the expression (4.34)
for &(h). Q
We are now in a position to prove
Theorem 4.4. Assume that (4.3), (4.4), (4.11) and (4.12) hold. Let u0
be given in H and f be given by
(4.36) / = /, + /„, UeS', UeL*(0,T;V).
There exists a unique wef such that
(4.37) A(t)u + u' =/,
(4.38) u(0) = u0.
(Note that (4.37) implies that u eY and u(0) is taken in the sense
of Theorem 4.3.)
Proof. Choose L as in Proposition 4.2. Then u eY and, by definition
of u(0) (see Theorem 4.3), we have
(4.39) [u(0),v(0)] = -(u',v)-(u,v') if v e 1T n D[A*', rT').
But according to (4.37):
(A (t) u, v) + « v) = (/, v) =!,»)- [«0f w(0)]
and with (4.39) (and since {A (t) u,v) = (u,A*(t) v)) we obtain
(u,A*{t)v - V) - [«(<>),!;(0)] = (L,v) - [uo,v(0)]t
from which we obtain
[«(0),i;(0)] = [uo,v{0)] V»6fnD(il*;f');
whence (4.38) (since v(0) describes H as v describes V n D{A*\ y^r/)).
For uniqueness, go back over the calculations. D
4.7 Examples
We give some examples of applications of the preceding theorems.
4.7.1 Example 1
Let Q be an arbitrary bounded open set in R", r its boundary (which
may be "arbitrarily irregular"); let Q be the cylinder
Q = Qx]0,T[, T<oo, and Z = rx]0,T[.
Take
" (Y du dv \
(4.40) a(t;utv) = V \\aij{xft) dx + c(x,t) u'v) dx,
i,jtij \ dxt dxj J
Q

244 4. Abstract Parabolic Equations, Initial Condition Problems (I)
where
(4.41) aueL^(Q),
[ there exists oc > 0 such that
(4 42) 1 n n
Z *uMMiJj£*IIA,|2, VA.eC, *e!2, <e]0J[,
I ij-l i=l
(4.43) ceL°°(<2).
Eventually changing u into eAf w, with sufficiently large A > 0, we may
always assume that
(4.44) c(x,t)^0, V*el2, te]0,T[.
Next, take
H = L2{Q), V = Hl0{Q).
Theorems 4.1 and 4.4 apply to the boundary value problem (of
Cauchy-Dirichlet type):
8u
(4.45) +Au = f in Q,
ot
(4.46) u = 0 on Z,
(4.47) u{x,0) = u0 in 12,
Wnere „ ^ , ^ v
" 3 / 3« \
4« = -- X -7— K,^—) + cu.
u=i dxt \ dxj J
Let us specify the results thus obtained. Take
/eL2(0,r; #-!(£)), u0eL2(Q).
Then, from Theorem 4.1 (and Remark 4.3), we deduce that there
exists a unique u in
L2(0,T}H10(Q))nC0(0>T>L2(D))>
satisfying (4.45) in the sense of distributions in Q, and (4.47) in L2 (12).
The boundary condition (4.46) on Z is contained in the fact that
ueL2(Q}T,Hl{Q))
(if 12 is regular, then see Chapter 4, Section 2). □
Let us now apply Theorem 4.4; then we may take / in a larger
space: in fact (see Proposition4.1), we may take / in the form
(4.48) f = fo+ — (Qfi)+f2,
ot
q defined by (4.23),
U eL2(0, T\ HlQ(Q))9 i = 0, 1, f2 eL2(0, T\ H'^Q)).

4.7 Examples 245
From Theorem 4.4, we deduce that if u0 e L2 (Q), then there exists a
unique u in
L2(0,T; #£(£)),
satisfying (4.45) in the sense of distributions in Q and (4.47) in the sense
of Theorem 4.3, condition (4.46) still being contained in
^ei2(o,:r;#£(£)). □
4.7.2 Example 2
Again, we choose Q and a{t\ u, v) as in 4.7.1; we may also assume,
eventually changing u into eAf u, A > 0 and sufficiently large, that
I there exists m > 0 such that
c(x,t) ^ m, VxeQ, /e]0,T[.
Let
H = L2(Q), V = Hl(Q).
"Formally", we obtain the problem (of Cauchy-Neumann type):
du
(4.45) +Au = f in Q,
du
(4.50) -— = 0 on Z,
dvA
(4.47) u(x,0) = u0 in Q,
rs
where is the "co-normal" (or "transversal") derivative with
dvA
respect to A,
(i.e. if Q is regular: = — Y aijcos{xjtv) )•
\ dvA ij=i dxj
We must call attention to the fact that in this case V = (H1 (Q))'
is not a space of distributions on Q and therefore L2(0, T; V) is not
a space of distributions onQ. In order to obtain the usual interpretations
we may introduce a space S1 (Q) (see Chapter 2, Section 6.3); but
since we have made no regularity assumptions on Q, this requires some
further developments.
The space S1 (Q).
We introduce
(4.51) 6{x) =mi(d{x,D,\)>
where
d(x, T) = distance from x to jT.

246 4. Abstract Parabolic Equations, Initial Condition Problems (I)
Note that d e L00 (12).
We define (analogously to Chapter 2, Section 6.3, but now d is not
infinitely differentiable):
dv
(4.52) Sl(Q) = "
We provide J?1 (12) with the norm
v\veL2{Q)}d^— eL2(Q),i= lf...,n}.
3^j
i»i2 + i
<5v
d#f
2x 1/2
In this manner we obtain a Hilbert space. We now extend
Proposition 6.2 of Chapter 2 to
Proposition 4.3. The space 2(Q) is dense in J?1 (12).
Proof. Define:
Ee = {x\xeQ,d(x,r) ^e},
%E = characteristic function of Ee,
cpE = function belonging to @(Rn), ^ 0,
of the form cpE(x) = e~n (pi — j,
<pe^(R"), y^O, j<p(x)dx=\,
(p with support in \x\ g \, and finally let
<5e = Xe*<Pe-
We have:
5E(*) =0, if d(x,D ^y
af(«) = i, if d(x,r) ^
3e
We immediately see that if u eS1(Q), deu -+ u in L2(D) as e -► 0.
To show that
(4.53)
eu -+ u in S1 (12) as e -► 0,

4.7 Examples 247
we still have to show that
d du
d — (dsu)-+d— in IS(Q)
oxt oxt
and for this purpose it is sufficient to show that
dd,
u -► 0 in L2 (Q).
dxit
But
ddF 3e
= 0 if d{x,r) ^ and d
dxt 2
dxt
w
is bounded, whence the result and (4.53).
To complete the proof it suffices (as for the case where the boundary
of Q is regular) to regularize deu, with fixed e. D
Thus, we define
(4.54) S-l (Q) = (S1 {£)))' (dual of S1 [£)));
E~1 (Q) is a space of distributions on Q.
According to Proposition 4.3 and the Hahn-Banach Theorem, we
have
Proposition 4.4. Every element f of S~l (Q) may be represented, non-
uniquely, by
/ = /o + I-^-Wi). /i6iaW, i = 0,...,n. U
1 = 1 OXt
Note that we have
(4.55) H1 (Q) c S1 {£)), S~' {£)) cz (H1 (Q))'
(Hl (Q) is dense in S1 (D), by Proposition 4.3, and therefore we may
identify S~l{Q) with a subspace of (H1{Q)y).
It follows that
(4.56) L2(0, T;t S-'(Q)) cz L2(0, T\ (i/1 (Q))')
and that L2(0, T',S~1(Q)) is a space of distributions on Q.
We come back to the Cauchy-Neumann problem and take
feL2(Q,T\E-l{Q)), u0 = 0
(we could also take u0 e L2 (Q), but we take u0 = 0 for the sake of
simplicity).

248 4. Abstract Parabolic Equations, Initial Condition Problems (I)
We may apply Theorem 4.1 and find a unique u in
L2(0, T\ H1 (£))) n C°(0, T\ L2 (£))),
with u(0) =0 and el2(0, T\ (i/1 (.Q))'), satisfying the equation
dt
T T
(4.57) \a(t\u{t),v{t))dt + L-^il,i;p) \dt
o
T
■I
= \<f(t),v(t)}dt, VveLtiO.TiH^Q)),
where [ , ] denotes the antiduality between (H1 (Q))' and H1 (Q) (and
the scalar product in L2 (D)) and < , ) denotes the duality between
S-^Q) and SY{Q).
It follows from (4.57) that u satisfies (4.45) in the sense of
distributions on Q.
Furthermore, we have the boundary condition (4.50) on E which is
contained in (4.57), but in a formal manner; indeed, formally
integrating by parts in (4.57), we obtain, thanks to the fact that v is arbitrary,
= 0 on 27.
But this is formal for two reasons:
(i) no regularity conditions have been imposed on 27 (and therefore
one may not speak of a "co-normal" to 27);
(ii) if we assume 27 to be regular, we must justify, in a certain sense,
the integrations by parts.
The justification of (ii) will be given in Chapter 4 of Volume 2; but
without regularity assumptions on 27, there exists (at the moment) no
du
other interpretation of the condition " = 0 on 27", other than
equation (4.57) itself. D dvA
Theorem 4.4 also applies to problem (4.45), (4.50), (4.47); this allows
us to take / in a more general space. Indeed, we may take / in the form
(4.58) / = /o + — (Q h) + f2 (Q defined by (4.23)),
or
with/.el^O.r;^1^)). » = 0,1, f2eL2(0,T;S-l(Q)).

4.7 Examples
249
Then, if furthermore u0 e L2 (Q), there exists a unique u in
L2(0, T)Hl{Q)) such that
T T T
a(t;u(t),v{t))dt - u(t),—- \dt = [f0(t),v(t)] dt -
0 0 0
T T
~\\Q[t)h[t)'~i?\dt+ \<**® •*&>*' +
(4.59)
0 0
+ [*o,v{0)], V^I^OJlFffl)), with
— eZ,2^,!;^1^))' and v(T)=0.
dt
It follows that u satisfies (4.45) in the sense of distributions on Q
and u(0) = u0 in the sense of Theorem 4.3; furthermore (4.59) contains,
in a formal way, the boundary condition (4.50) (considerations analogous
to those made for equation (4.57) hold). □
4.7.3 Example 3
Consider a bounded open set in R", with (suitably) regular boundary
J7 and assume that the subset ro of J7 is an (n — 1)-dimensional variety
with boundary (therefore ro + r)', 7\ = T — T0. Again, take a(t',u,v)
defined by (4.40) with (4.41), (4.42), (4.43), (4.49) (we could also
replace (4.49) with (4.44)). Finally, take
(4.60) H = L2{Q), V = {v | v e H1 [Q) ,v = 0 on T0}((1)).
The general theory applies again; "formally", the problem is (of
Cauchy type with mixed conditions on Z)\
(4.45) -?- + Au = f in Q
01
(4.61)
(4.47)
u = 0 on Z0 = r0 x ]0, T[,
du
dvA
= 0 on 2\ = rx x ]0, T[$
u(x, 0) = u0 in Q.
For the interpretation of the problem, we meet the same difficulties
as in 4.7.2 concerning the space V. We could use the space 3~1(Q),
but we can do better in the following way.
((D) p being assumed sufficiently regular, we may define y0 v if v € H1 (Q)
(Chapter 1, Section 8); we impose oni/GF that the restriction of y0 v to TQ vanishes.

250 4. Abstract Parabolic Equations, Initial Condition Problems (I)
The space Sxro (Q).
We introduce
(4.62)
and
dQ(x) =mi(d(x,ri),\)>
dv
(4.63) SlQ{Q) = \v\veL2(Q),d0-—eL2(Q),i = 1 »
Provided with the norm
dxt
dv
2\ 1/2
dxt
Sr0(Q) is a Hilbert space.
Let
rg = {x | x er0, d{x, dr0) ^p>0,dro= boundary of T0 in T}
and let 08 be an open set in R" contained in Q, a subset of whose boundary
d@p is ro, the rest of d@p being contained in Q\ then if v e5rQ{Q),
we have: veH1^^ and we can define
v\re, for all V/3 > 0.
Therefore we can define v\ro (in particular eL2(ro)).
Now, we can define
(4.64)
Blro(Q) = {v\veElro(Q),v = 0onr0},
which is a Hilbert space for the norm induced by the norm of Sp0 (Q).
Of course, we have
(4.65)
V defined by (4.60), S1 (Q) defined by (4.52).
We verify — by the same type of proof as for Proposition 4.3 — that
(4.66) 3(Q) is dense in 3\.a(Q).
We introduce
(4.67) Sr01(Q)=(S1ro(Q)).
Every element / of S^1 (Q) may be represented, non-uniquely, by
f = to + t-T-(»ott), (ftsL2(Q), » = 0 n). D
» = 1 OXi

4.7 Examples 251
Now we may use the space S^1 (Q) for problem (4.45), (4.61), (4.62),
(4.47), with u0eL2{Q); if we take
we can apply Theorem 4.1.
If, more generally, we take
d
f = fo+ — (Qfi)+ fi
ot
with U e£2(0, T\ V), i = 0, 1, f2 el2(0, T;^1 (£)), we can apply
Theorem 4.4.
The interpretation of the results is completely analogous to the one
given for the Cauchy-Dirichlet and Cauchy-Neumann problems. Note
that condition (4.61) will be expressed by the fact that, in both cases,
u belongs to L2{Q,T\V), with V given by (4.60); condition (4.62)
will be verified in a formal way; in the first case it will be contained
in an equation like (4.57) and in the second case in an equation like
(4.59). □
4.7.4 Example 4
Examples 1, 2 and 3 may be generalized to the case of operators A
of order 2m, m > 1.
Consider a form of the type
(4.68) a{t;u,v)= £ \ a„{x,t) D*uWv dx
and take
(4.69) H = L2 (Q), Ho{Q)cVc Hm (Q),
V closed subspace of Hm{Q).
We assume that the form a{t\ u, v) satisfies (4.3) and (4.4) so that
we can apply Theorems 4.1 and 4.4. Also, applying the results of
Sections 9.5, 9.6, 9.7 of Chapter 2, we are able to interpret the results in
the same way as for examples 1, 2 and 3.
Formally, we solve boundary value problems of the type
(4r+ I {-\)^D»{am{x,t)D*u) = t in Q
J ^ |P|.|fl|^m
| u(x, 0) = u0
I with certain boundary conditions on 27, which depend on V.
But we shall not go into the details of this type of application of the
theory here. □

252 4. Abstract Parabolic Equations, Initial Condition Problems (I)
4.7.5 Example 5
The "general" choice of H for the applications is L2(Q). But this
must not necessarily be the case; see Chapter 2, Section 9.9. We shall
give an example of this kind here. We take
(4.70) H = Hl0{Q),
with
"• f du dv
(4.71) («.»)n = I \-7-T-dx>
i = i J dxt dxt
Q
in this way we define a hilhertian scalar product on H, assuming Q to
be bounded.
Further, we take
(4.72) V = \v\veH, Av eL2{Q), i = 1,...,»}
I dxt J
and
n II d 1
(4.73) \\v\\2v= |M|J + I \\-r-Av\
1 = 1 II ox(
2
\L2(Q)
Finally, for all u,v eV, let
^ H d \ / d \
(4.74) a(u,v) =£ 1 [j^AuJ ['fo-Af})dx'
Q
Of course, we have
(4.75) a{v,v) + X\\v\\2H^mm{\,?.)\\v\\* (VA > 0), V^eF,
and consequently, according to Theorem 4.1, we have:
Proposition 4.5. Let H and V be defined by (4.70) and (4.72). Let V
be the antidual of V when H is identified with its antidual. Let f be given,
with
(4.76) feL2(0,T;V).
Then, if u0 is given in H, there exists a unique u e L2(0,T',V), such
that, a.e. in [0, T],
(4.77) a(u(t),v) + [uf(t),v] = [f{t),v]t Vv e V,
(4.78) u'eL2{0,T;V),
(4.79) «(0) =«0,
where the scalar product [/ (t), v] denotes the duality between V'
and V. 0
Remark 4.4. We must be careful with the interpretation of V\ □

4.7 Examples 253
In particular, let us take
(4.80) feL2(0,T;H) (i.e. / eL2(0, T; Hl0(Q)))
in (4.77).
Then (4.77) may be written:
(4.81) a(u (t),») + —[«(t) .»] = [/(0, »)], V«eF.
^ at
But
[/W, »] = (/W, -^vW), [«(*), »] = (u(t), -Ji;)L2(0).
If, ^cpy^eH1{Q)) we set:
(4.82) &(9m0=I -t1-^-^'
i=i J oxi dxt
then fi
a(u(t),v) = b(-Au(t), -Av)
and (4.81) may be written:
(4.83) d
b(-Au{t),w) + —(u{t),w)L2(Q) = (f{t),w)L2(Q), Vw = -Av, v e V.
at
But, as v describes V, Av describes a space W containing H1 (12) and
therefore (4.83) holds ^w e H1(Q). But then it follows (see Chapter 2,
Section 9.9) that u satisfies
du
(4.84) + A2u = / in Q, u{x, 0) = u0 in 12,
dt
dAu
(4.85) « = 0, = 0 on Z.
dv
Remark 4.5. The operator —A is the infinitesimal generator of an
analytic semi-group in H, see Yosida [2]. D
Remark 4.6. Problem (4.84), with the boundary conditions
du dAu
(4.86) ir = 0> -3— = ° on^
can be treated in the same way. D
4.7.6 Example 6
Let 12 be an open set in R2. Take
v | v eL2(12), — e L2(12), —4"e L'(fi)l»
ox^ dx2 J
f du ~dv f 32w 32v
(4.88) a(*; «,»)=— —-<** + —T -r-rdx.
j ox1 oxx J ox2 ox2

254 4. Abstract Parabolic Equations, Initial Condition Problems (I)
The general theory applies. For the interpretation of the problem,
one possibility is to introduce (see Chapter 2, Section 6.3 for analogous
considerations) a space K(Q) such that
(4.89)
f V czK{Q) czL2{Q)
9{Q) is dense in K{Q)
I K(ii) is "the smallest possible".
We shall make the hypothesis that Q is bounded and has a regular
dv
boundary. Then every v e V also verifies e L2 (Q). We define d
by (4.51) and K{Q) by d%2
dv dv d2v
oxx ox2 ox2
We provide K(Q) with the norm
(4.90) K(Q) = \v\veL2(Q),d-^— ,d^— ,d2^-TeL2{Q) .
■> II dv |r || dv
\v\\han + M-— +
I2 II d2v II2 y'2
L2(Q) OX2 \\L2(Q)/
which makes it a Hilbert space.
With the same kind of proof as for Proposition 4.3, we verify that
2(Q) is dense in K(Q).
Using K(Q) in an analogous way as S1(Q) in 4.7.2, we may apply
the general theory; in particular, taking
feL2(0,T;K'{Q)), uQeL2{Q),
we can apply Theorem 4.1; we obtain u e L2 (0, T\ V) n C°(0, T; L2 (Q))
such that
du d2u d*u
=~ H t- . = / in the sense of distributions on Q,
dt d%i dx2
[ u(x,0) = u0.
Thanks to the fact that u e L2(0,T,V), the solution also verifies
certain boundary conditions on £, which, for example when Q is the
rectangle
Q = ]0,a1[x]0,a2[,
are "formally" given by
u = 0, on the sides of J7 parallel to the #2-axis, W e]0, T[,
du
u = = 0 on the sides of r parallel to the #raxis, V£ e]0, T[.
dXy

5.1 Some Interpolation Results
255
Remark 4.7. For non-homogeneous parabolic problems treated from
the "variational" point of view, the reader may also consult Lions [13],
Chapter 6, Section 9. □
5. Example: Abstract Parabolic Equations,
Initial Condition Problems (II)
5.1 Some Interpolation Results
Generally, if F is a Hilbert space and (9 an open set in Rf, we define
(5.1) Hl(&;F) = {f\fsL2(&;F),f'sL2(0;F)}
which is a Hilbert space for the norm
lj(\\f(t)\\2F+\\f'(t)\\2F)dt
Also, we recall (see Chapter 1, Section 7.1 and Remark 9.5) that
#1/2(R;F) = {/|(1 + |r|)1/2/eZ2(Ri;F)}
(where / = Fourier transform in t of /).
We have (see Remark 9.4, Chapter 1):
Proposition 5.1.
(5.2) [Hi(R; V'),L2{R't V)]1/2 = H^(R',H).
Proof. We consider V as the domain of a positive, self-adjoint,
unbounded operator in H, which we diagonalize (as in Chapter 1,
Section 2.1) by a unitary operator °ll. Then if °ll H = f) and if we apply
the Fourier transform in t and the operator % to L2(R, V) (resp.
#*(R; 7')), we see that L2(R, V) (resp. #X(R, V')) is isomorphically
transformed to the space of /'s such that
(5.3) A/eL2(R;I))
(resp.
(5.4) (l + |T|)y/eZ2(R;I))).
Then, by definition of the spaces [X,Y] (Chapter 1, Section 2.1),
we see that [H1(R] V'),L2(R] V)]l/2 is isomorphically transformed to
the space of /'s such that
(5.5) A1'2 (1 + |r|)1/2 -^Z = (1 + M)1'2 / e I2 (R; D)
which is the transformed space of H1,2(R\ H). □
1/2

256 5. Abstract Parabolic Equations, Initial Condition Problems (II)
Now, we define
(5.6) oW(0,T;F) = {v\veHi(]0,T[-F) (see (5.1)), v(0) = 0},
closed vector subspace of H1 (]0f T[; F), provided with the induced
norm. Further, we define
(5.7) tf1/2(0, T\ F) = [ ffi(0, T\ F),L2(0, T; F)]1/2>
(5.8) oH^(0,T)F) = [QHH09T\F),L*$9T\F)-\ll2.
Then
Proposition 5.2. W^ have
(i) tf1/2(0, T; i7) = space of restrictions to ]0, T[ of tf1/2(R; F);
0H1,2(0, T]F) = space of restrictions to ]0, T[ of the elements
of H1/2(R; F) which vanish for all t < 0.
Proof. Result (i) is verified exactly as in Chapter 1, Theorem 9.1.
For (ii), we shall show that
0H1/2 (0,T,F) = space of restrictions to ]0, T[ of the elements
of H1/2( — co, T\ F) which vanish for t < 0;
then we obtain the result by applying (i), in which we can replace 0
by -oo.
To show (5.9), we first consider the mapping u -► u = extension of u
by 0 for t < 0, which is a continuous linear mapping of
oH1(0,T,F)-+H1(-<x>,T;F) and of L2{0tT\F)-+L2{-oz,T)F),
therefore, by interpolation, of
0tf1/2 (0, T; F) -+ H^i-oo ,T\F).
On the other hand
v -> r v, rv(t)=v(t)— v( — t),
is a continuous linear mapping of
H1(-od>T',F)-+0H1(0,T;F), Z2(-oo, T\ F) - L2(0, T\ F) t
therefore, by interpolation, of tf1/2(-oo, T\ F) -► 0#1/2(0, T\ F).
Since r u = u, the desired result follows. D
Proposition 5.3. We have:
(i) [H1(0,T;V'),L2(0,T;V)]ll2= H^(0,T;H),
(ii) [^(0, T; V'),L2(0, T; V)]1/2 = 0H^(0, T;H).
(of course, as usual, with equivalent norms).
(5.9)

5.2 Interpretation of the Spaces &1'2 and &%2 257
Proof. The same proof as for Proposition 5.2 shows that
[off^O, T; V), L2{0, T) V)]1/2 = space of restrictions to ]0, T[ of
the elements of
[H1 (-co, T;V'),L2(-co, T;V)]1/2
which vanish for t < 0 and that
[H1( — oo, T\ V), L2(-co, T; V)]1/2 = space of restrictions to
]-oo, T[ of the elements of [tf^R, F),Z,2(R; 7)]1/2.
From which, with Proposition 5.1:
[o^MO, r; F),Z,2(0, T; 7)]1/2 = space of restrictions to ]0, T[
of the elements of i/1/2(R; #) which vanish for t < 0, whence (ii) (and
(i) with obvious modifications), thanks to Proposition (5.2) (ii). □
Proposition 5.4. Every element u of 0H1/2(0, T;F) satisfies
T
(5.10) JY1 \\u{t)\\2Fdt< 00.
0
Proof. Same as for Theorem 11.7 and Remark 11.5 of Chapter 1. □
5.2 Interpretation of the Spaces <P1/2 and <P*/2
We now apply the theory of Sections 3.2 and 3.3 to the setting of
Section 4. In particular:
f d
(5 11} I un^er the hypotheses of Theorem 4.1, A(t) + — is an
I isomorphism of 01/2 onto ($i/2)',
where
0l'2 = rn[D(A*;r'),r]1/2.
Therefore, in the notation of Section 5.1, we have
01'2 = L2(0, T; V) n [oHi(0, T; F),L2(0, T\ V)]lj2.
Applying Proposition 5.3 (ii), we obtain:
( 01/2 =L2{0, T;V)n0Hll2{0, T\H).
(5.12)
I 0#1/2(0, T\ H) defined by (5.8) or Proposition 5.2(ii). □
Now, we set:
j THl(0,T;F) = {v\veH1(0,T;F),v(T) = 0};
(fU3) ifT<oo.

258 6. Abstract Parabolic Equations, Periodic Solutions
Then (interchanging the roles of 0 and T):
(5.14) 0'J2 = L2(0, T] V) n r#1/2(0, T\ H)
(eliminating the index T if T = +oo). □
Remark 5.1. It follows from Proposition 5.4 that the spaces 01/2
and 0]J2 are distinct, since u e 01/2 satisfies (5.10), where F = H,
whereas u e 0]J2 satisfies
T
j(T - Q-1|«WIh^< oo. □
0
Remark 5.2, For ^ 6 01/2, we have (see Chapter 1, Theorem 4.1)
D^ueL2(0,T'tH)t
D]!* being defined by Fourier transform after extension to Rf. □
Remark 5.3. If, instead of ]0, T[, we consider Rf, taking for G(s)
the semi-group (in fact the group!) of translations on R0, we satisfy the
hypotheses of Theorem 3.2; in this case 01/2 = 0]J2. □
Remark 5.4. The results of this section apply to the examples given
in Section 4.7. □
6. Example: Abstract Parabolic Equations, Periodic Solutions
6.1 Notation. The Operator A
We consider the setting of Section 4.
The spaces V, Jtf, if' are the same as in Section 4, but this time
we always have T < oo. The operator M is defined as in Section 4 (4.7).
On the other hand, the semi-group G (s) is 'different than the one in
Section 4.3.
Here, we define:
\f(t-s + T), if 0<*<s
(6.D G(s)f(t)=\
[f(t - s), if s < t < T.
Then, we have
dv-
(6.2) Av = — = v'
with
(6.3) D(A;-r') = \v\ver',— = v'er',v{0) =v{T)\,
with analogous descriptions for D(A; Jt) and D(A;'f).

6.3 Choice of L
259
The semi-group G(s) (in fact the group) is a contraction semi-group
(in fact unitary) and we may apply the general theory. □
6.2 Application of the Isomorphism Theorems
Theorem 1.1 yields:
Theorem 6.1. Assume that (4.3) —(4.4) hold. Then, for f given in
V = L2(0, T; V')} there exists a unique wef such that
(6.4) A [t)u + u' = /
(6.5) u{0) = u(T).
(As in Theorem 4.1, (6.4) implies that u' eZ,2(0, T\ V) and then (6.5)
has meaning according to Theorem 3.2'of Chapter 1.)
Proof. Choose A as in (6.2), (6.3) and note that if u e *T n D (A) IT'),
we have (6.5). D
Remark 6.1. Condition (6.5) is a periodicity condition: if / is given
on R, takes its values in V and has period T and if A(t) is defined
on R, with period T, we deduce the existence of a (unique) solution
with period T from Theorem 6.1. □
Theorem 2.1 yields:
Theorem 6.2. Assume that (4.3) —(4.4) hold. For L given in (V* r\
n D(A] y'))', there exists a unique ueV such that
(6.6) (u,A*{t)v -vf) = {L,v), Vver nD(4*;f).
Proof. Application of Theorem 2.1 and of the fact that, A being
defined by (6.2), (6.3), we have
(6.7) A*v = -v'f
(6.8) D{A*',r') =D(A\r'). a
6.3 Choice of L
Let qx (compare with (4.23)) be defined by:
T
(6.9) Ql(t)=\
t!t0, if 0 <; t ^ t0t fixed t0 <
1, if t0^t ^T - t0>
T - t
, , if T - t0 < t < T.

260 6. Abstract Parabolic Equations, Periodic Solutions
Next, we define Ex (compare with (4.24)) by:
(6.10) St = {v\veL2(0,T;V) = r1, Qlv' ef'},
a Hilbert space for the norm
(6.H) (IN^+lleif'llr.)172.
We have (as in Section 4):
Proposition 6.1. The space ^(]0, T[\ V) is dense in Ex.
Every f# e E[ may be written — non-uniquely — as
(6.12) /* = /„+ -jr(e*/i). fteL2(0,T;V). Q
at
Then we choose L in (6.6) by (compare with (4.26)):
j (L, v) = U, v) + (/„, i;) + [uo,v(0)]
[UeSi, U*e-r', u0eH. D
6.4 Interpretation of the Problem
Theorem 6.3. Hypothesis of Theorem 6.1. Let f and u0 be given
(6.14) / = /* + /**, f*eEi, Uerf, u0eH.
There exists a unique u e'V such that
(6.15) A{t)u + u' =/,
(6.16] -u{0) + u{T) =u0,
where u(T) — u (0) is taken (as in Theorem 4.3) by extension by continuity.
We only give the
Outline of the proof.
First, wfe verify (as in Theorem 4.4) that u, solution of (6.6) with
the choice (6.13) for L, satisfies (6.15).
Then ueYlf where this time (compare with (4.28)):
(6.17) Yx = {v\veiT,v'eE[ + f'}.
As in Theorem 4.3, we show that @([0,T]',V) is dense in YX
and that the mapping
u^{-u(0) + u(T)}t
of
9([0,T]',V)^V,
extends by continuity to a mapping, still denoted u -► { — u (0) + u (T)},
of Yx -+ H.
Then (6.16) follows by integrations by parts, as in Theorem 4.4. D

7.1 The Elliptic Problem 261
6.5 The Isomorphism of <P1/2 onto its Dual
According to (6.8), we may apply Theorem 3.2, therefore:
Theorem 6.4. Hypothesis of Theorem 6.1. Then A(t) -\ is an
isomorphism of 0i/2 onto its dual (0l/2)'. □
There remains to interpret 01/2.
Proposition 6.2. The space &1/2 = V n [D(A] "T'), ^]1/2 is the
space of functions u:
t? / int\
(6.18) u = 2, un exp I 2tz^t- I> «,, e 7,
such that
(6-19) IHI = (l [II«Jk + (1 + I»[)II«.I|J]) <<x>.
(||| m 1 being a norm equivalent to the initial norm).
This result is a consequence of
Proposition 6.3. We have
(6.20) [D(A; i^'), f]1/2 = space of u's of the form (6.18),
with
(+f U + I*I)KIIh)
1/2
< 00.
Proof. The idea is the same as for the proof of Proposition 5.1: we
consider D(A] "T') (resp. f) as domain of Jl (resp. /2), a positive self-
adjoint operator in V. The operator Jl is given by (here we replace
the Fourier transform by the Fourier series):
(6*21^ ^ 12n\nt\ ^ (2n\nt\
/i/ = I(l + l»l)/-exp^—^-j, if / = £/.exp(—^-j
and the operator J2 is given by the diagonalization operator.
The operators Jt and J2 are commutative and we use Section 13 of
Chapter 1. □
7. Elliptic Regularization
7.1 The Elliptic Problem
In this section, we shall give a new proof of Theorem 1.1 by the
so-called "elliptic regularization method", which consists in
"approaching" the operator A + M with a family of operators of a more "elliptic"
nature.

262 7. Elliptic Regularization
Theorem 7.1. Let e > 0 be fixed. Under the hypotheses of Theorem 1.1,
there exists a unique ueeir c\D{A) £P) such that
(7.1) {Aue,v) + {Mue,v)+e(AuE,Av) = {f,v)> Vv ef c\D{A\tf).
Proof. For u, v e tT n Z)(/l; Jf), set
(7.2) ne(u,v) = {Au,v) + [Mu,v) + e{Au,Av).
We have
ReIIE{v,v) = Re(^v,i;) + Re{Mv,v) + e \\Av\\2^
and, according to (1.5) —(1.9), we therefore have
(7.3) Rene(v,v)^oc\\v\\2ir + s\\Av\\23e,
whence the result, by Theorem 9.1 of Chapter 2. □
Remark 7.1. (7.1) is called the "regularized elliptic problem" for
problem (1.10). □
Let us now verify the following estimates:
Proposition 7.1. For e > 0, we have
(7.4) Kll^c,
(7.5) >fi\\AuJ„£c,
(7.6) AugeD(A*\r'),
(7.7) \\Aut\\r.£c.
Proof. (7.4) and (7.5) follow from (7.3).
(7.1) may be written in the form:
e{Aue,Av) = (/ - Aue- Mue,v), v eiT n D(A',JP).
and we see that v -+ (Aue> Av) is continuous on the space
D[A;ir)[c r nD{A',jr))
in the topology induced by "T and therefore (Lemma 1.3) we have (7.6)
and
e(AuE,Av) = e(A* Aue,v),
from which we obtain
(7.8) eA*Aue + Aue + MuE = f.
Therefore, since, for e > 0, the operator e A* + I is invertible in
&{ir'9ir'):
Aue = {eA* + I)-1 (/- MuE)

7.2 Passage to the Limit 263
and therefore
(7.9) \\Aue\\r. g \\(sA* + /)-MI«*-^o 11/ -MuJr..
But, according to (7.4), ||/ — Mub\\y* is bounded and since — A*
is the infinitesimal generator of a bounded semi-group in ^',
&ar>'ir') = constant,
we deduce (7.7) from (7.9) and (7.4). □
7.2 Passage to the Limit
Theorem 7.2. Assume that (1.5) and (1.9) hold. Let ue be the solution
of the "regularized elliptic problem" (7.1). Then, as e -* 0, we have:
(7.10) ue-+u in f,
(7.11) AuE-+Au in ->r',
z^re w is /Ae solution of (1.10).
Remark 7.2. The proof which follows agam yi^s /Ae existence of u
satisfying (1.10); the uniqueness must, at any rate, be shown separately
as in 1) of Section 1.4. □
Proof of Theorem 7.2. From the estimates (7.4) and (7.7) it follows
that we can extract uv, tj -► 0, so that
un -> w weakly in y,
Aun -+ % weakly in "T''.
But since A is closed in "T*', we see that
(7.13) weD(A;iT'), x = Aw.
Now, Ae&{p[A\ir)\ir) yields, by transposition,
(7.14) A*e&{r''t{D{A\r))')
and therefore
A* A un-+ A* Aw weakly in (D {A; *"))',
and consequently
f/A*Au1l-+0 in D(A;r')y
and (7.8) (for e = rj) yields
4w + Mw = /, K/efn/)(/l;f')
in the limit.
Therefore w — u and we have
(7.15) ue-+ u weakly in i^, Aue-+ Au weakly in V.
(7.12)

264 7. Elliptic Regularization
There remains to show the strong convergence. For this purpose,
let
pE = (M(ue — u), ue — u) + (A (ue — u), ue — u) + e(A ue, A ue).
According to (7.1) and since Au + Mu = /:
Pe = (/»«0 - (/. «,-«)- (M «f + ^1 «if «) -► 0, as e -> 0.
But
Re£E ^(x\\ue- u\\y,
whence (7.10).
Finally
A ue = (e /I* + I)"1 (/ - M uE) -+ f - M u
strongly in V, whence the theorem. D
Remark 7.3. In the setting of Section 4, if for example A (t) = — Ax
(Laplacian); the "initial'' (non-regularized) problem being for example:
- du
—Axu H = /, x eiJ, an open set in R", t e ]0, T\,
dt
(7-16)| u(x,0)=0
I u = 0, for xsdQ = boundary of Q, te]0,T[,
the "regularized elliptic" problem is
(7.17)
Axue +
dt
d2u.
— s
= /
dt2
ue{x,0) = 0
dt
(x, T) = 0
ue = 0 for %ed£, *e]0,r[,
which is indeed a (mixed) elliptic problem.
If A is a differential operator of order 2m, we can replace
or 6y
.4 h by ^4 + e
a d2m
A + + (-ire——;
dt dt2m
similarly, in the general theory, the term e(A ue, A v) in (7.1) may be
replaced by s(Am ue, Amv), for example. D

8.2 Existence and Uniqueness Theorem
265
8. Equations of the Second Order in t
8.1 Notation
Let V and H be defined as in Section 4.1. Let a(t\u9v) be a family
of continuous sesquilinear forms on V, such that
(the function t -* a(t;u,v) is, \/u,veV, once continuously
differentiable in [0, T], with T < oo, for example,
I a(t\ u, v) = a(t\ u, v), V/*, ^eF, and for a suitable A:
j a(t\v,v) + AM2 ^a|M|2, a > 0, V^eF.
Remark 8.1. The hermitian symmetry hypothesis: a(t;u,v) = a(t;u,v)
can be generalized to the case for which only the "principal part" of
a(t',u,v) is hermitian. See, for example, Lions [13]. □
Let A (t) e HP(V', V) be the operator defined by a(t',u,v) as in
(4.5), (4.6).
We consider the equations
2
(8.3) A(t)u(t) + u"(t)=f(t), u"=—u,
dtz
with the Cauchy data:
f u (0) = u0,
(8.4)
( «'(0) = «!• □
8.2 Existence and Uniqueness Theorem
Theorem 8.1. Assume that (8.1) and (8.2) hold. Let /, uQt ux be given
with
(8.5) /eL2(0,r;i/)
(8.6) ^0eF, %e#.
Then there exists a unique function u satisfying (8.3), (8.4) and
(8.7) u eL2(0,T;V),
(8.8) u' eL2{0,T;H).
Remark 8.2. If (8.7) holds, then A (t) ueL2{0, T\ V), so that (8.3)
implies
(8.9) u"eL2(0,T',V).
Then u(0) and u'(0) are well-defined, so that (8.4) has meaning. □
The proof of Theorem 8.1 proceeds in two stages. In the first part,
energy inequalities are established (see (8.15), below); in the second part,

266 8. Equations of the Second Order in t
we show how to use these inequalities to obtain the existence of a
solution (we give one method: Faedo-Galerkin, see Faedo [1] (and also
Green [1]); there are others, see Lions [13] and the Comments; and
also Section 8.4).
Uniqueness is treated separately; if the solution u is "regular",
uniqueness is immediate. □
First part. A priori estimates.
Equation (8.3) is equivalent to
(8.10) a(t',u[t),v) + [u"(t),v] = \f{t),v], VveV.
Formally taking "v = u'[t)" in (8.10) and taking twice the real
part of the equality thus obtained, we obtain (using the hermitian
symmetry of a(t]u,v)):
(8.11) d
a(t\u{t),u'(t)) + a{t;u'[t),u{t)) + \u'{t)\2 = 2Re[/(/), u'{t)].
dt
Set
d
a!(t; u,v) = a(t; u, v), Vu, v e V)
dt
(8.11) is equivalent to
(8.12) —(*(*; *{t),u{t)) + \u'(t)\2) - a'(t; u{t)tu{t)) = 2 Re [/(*), W (t)}
at
and integrating from 0 to t, we obtain:
(8.13) a(t;u(t)tu{t)) + \u'{t)\2 = a{0;uo,uo) +
T T
+ Kl2 + ja'(a;u(a),u(a))da + 2Re f [f(a), u'(a)] da.
o o
The second term of (8.13) is less than or equal to (the c's denoting
various constants):
T T
c\\u0\\2 + |%|2 + c|||^((r)||2^(r + 2j|/((r)||^((r)|^(r
0 0
and the first term is ^ oc \\u{t)\\2 + \u'{t)\2. Therefore:
(8.14) ||^)||2 + \u>(t)\2 :gcJK||2 + \ux\2 + j\f(a)\2da\ +
T
+ cj(\\u(<j)\\* + \u'(o)\2)da.

8.2 Existence and Uniqueness Theorem 267
Applying Gronwall's lemma, it follows that
(8.15) ||«WII2 + |«'WI2^
^cl\\u0\\2 + Kl2 + j\f(<y)\2d<y\, o^t^T. n
Second part. Proof of the existence.
In order to slightly simplify matters (but there is nothing essential
to this), let us assume V to be separable. Let wlf . . ., wm> . . . form a
"basis" for V in the following sense:
— for all m, wlt . . ., wm are linearly independent;
— the combinations £ £iwit ^eC, are dense in V.
finite
(Such a basis always exists if V is separable.)
We define the "approximate solution" um(t) of order m of the problem
in the following way:
m
(8.16) «.W = Igi-W»i.
i = l
the gjm(<)'s being determined so that
(8.17) a{t; um(t),Wj) + [<(t), wj\ = [/(<).»,]. 1 ^ / g m,
(8.18) gta(0)=fta, g;m(0)=J?(m
with ,
Yj^imwi~* uo in F as «->oo,
(8-19)
X *?*m wi -> ui in # as ra -► oo.
Thus, the gim's are determined by a fow^r differential system which
admits a unique solution. The same calculations as in the first part
show that
(8.20) ll^mWll2 + l^mWI2 ^ constant independent of m.
Therefore, in particular (and it may be seen that the result cant be
improved upon; see Section 8.4):
!um (resp. u'm) remains in a bounded set of L2(0, T\ V)
(resp. L2(0,T)H))
and we may therefore extract u^ from um so that
Up-* u weakly in L2(0,T] V)
u'n ~+ X weakly in L2 (0, T; H).

268 8. Equations of the Second Order in t
But x = u'
There remains to be shown that the function u constructed in this
manner is a solution of the problem.
To this end, we introduce
Ct[0, T] = {<p\<pe Cl([0, T)],<p(T) = <p'(T) = 0}
and consider the functions
(8.22) V = 5>j®w,.
j = i
For m = fi > fi0> we deduce from (8.17) (by multiplying by q>j(t)
and summing over / from 1 to /u0) that
T T
/[«(';%>v) - K>^~\dt = {[/• v]dt + W), v(0)].
0 0
It follows from this and from the second condition in (8.19) that
(8.23)
f{a{t\u,v) - [u',V'])dt = j\f,V]dt+ [u,v{0)],
0 0
V^ of the form (8.22).
But since the ww's form a basis for V, the set of functions of the
form (8.22) is dense in the space of functions ip e L2 (0, T\ V) such that
ip' eL2(0,T;H), ip{T) =0.
It follows from this that u is a solution of the problem. □
Remark 8.2. The preceding proof yields, without further effort, a
more precise result than the one in the theorem: it follows from (8.20)
that
!um (resp. u'm) remains in a bounded set of L°°(0, T; V)
(resp. L"(0,T;H)),
from which it follows that
ithe solution u of problem (8.3), (8.4), (8.7), (8.8) satisfies
ueLco{0tT'>V)> u'eL*(0,T\H).
This result will be improved upon in Section 8.4. □
Proof of uniqueness in Theorem 8.1. Let se]0, T[. Set
y>(t) = { J
0 , t^s,

8.2 Existence and Uniqueness Theorem 269
where u is a solution of (8.3) - (8.4) (with (8.7) - (8.8)) for / = 0, u0 = 0,
ux = 0.
Then, obviously,
T
f [A (t) u + u", y>]dt = 0
o
and by integration by parts (permissible), we obtain
T
[a(t\ u, \p) — [u't xp']~\ dt = 0
or
\[a{t,\p',\p) - [u'tu\\dt = 0,
o
and thus
o o
from which
a(0; v(0). y(0)) + N*)|2 = J <*'(*; V; y) dt-
o
It follows that
IIV(0)||2 + Ms)\2 g c, M \\f(t)\\2dt + \f(0)A.
But, if we set
t
w[t) = j «(cr) rfcr,
o
we may write this last inequality:
\\w(s)\\2 + \u(s)\2^c1ij\\w(t)-w(s)\\2dt+\w(s)A,
whence
(1 -2clS) \\w(s)\\2 + \u(s)\2 g c2j(\\w(t)\\2 + \u(t)\2)dt.
0
Choose 50 with, for example, (1 — 2cx s0) = \.

270 8. Equations of the Second Order in /
Then, for s -^ s0, we have
\\w(s)\\2 + \u(s)\2 ^ c3j(\\w(t)\\2 + \u(t)\2)dt
0
therefore u = 0 in [0, s0].
The length of s0 being independent of the choice of the origin, it
follows that u = 0 in [s0, 2s0] and so on, whence the uniqueness. D
8.3 Remarks on the Application of the General Theory of Section 1
8.3.1 "Vector" Notation
We shall investigate how the general method of Section 1 may be
applied to the problem
f A{t)u + u" =/,
(8.26)
I u{0) = u'{0) =0.
Setting u = u1, = uz, and introducing
dt
we may
(8.27)
write (8.26)
du
in
f 0
u = [u1
the form
1 u =
0 o
u2},
{0,/
Replacing u by exp (k t) u and setting g = exp ( — k t) /, system (8.27)
is equivalent to
du (kl -I\
<8-28> «-+U *i)a-{0'8)'
We choose (which is permissible):
r k = X + fi,
(8.29) ax{t\v,v) = a{t\v,v) + X[v]2 ^oc\\v\\2 + $[v]2 V^eF,
[ 2pax(t; v, v) ^ a'x(t\ v, v) = a!' (t\ v, v) Vv e V.
Then, we introduce
A d II 0\ d
III -1
(8.31) M =

8.3 Remarks on the Application of the General Theory of Section 1 271
and (8.28) may be written
(8.32) Au + Mu = {0,g\.
As we shall see in Section 8.3.2 below, the theory of Section 1 does
not apply to (8.32), but we shall indicate a simple " regular ization" which
allows the application of Section L
8.3.2 Regularization
For e > 0, we introduce
IXI -I \
(8.33) MF =
V ' £ \A(t) e(A(t) + XI) +kl)
and we show that the theory of Section 1 applies to
(8.34) AuE + Meue = {0tg}.
(It will then be easy to let e -> 0.) D
The space 34?.
We define:
(8.35) Jf = {zJl^ei^O^iF),?;^!2^^;^)}
provided with the scalar product (thanks to the symmetry of a(t\u,v),
it is indeed a scalar product):
T
(8.36) {*,€) = j(ax{t;u1,v1) + [u2,v2])dt
o
(where ax(t; u1, v1) = a(t; u1, v1) + X[u\ v1]).
Thanks to the second hypothesis in (8.29), this scalar product is
equivalent to the "natural" scalar product:
T
\((u\vi)v + [u*tv*])dt.
0
The space Y°\
(8.37) r = {v | v1 eZ,2(0, T\ V)tv2eL2(0t T\ V)},
provided with the scalar product
T
(3.0)^ = J («*(*; f*1.*1) + (u2,v2)v)dt.
0
The space Y°' is characterized by
(8.38) *T = {vlv1 eL2(0,T',V),v2eL2(0,T;V')}. D

272 8. Equations of the Second Order in t
The operator Me.
If v e ir, we have
Msv = {X v1 - v2, A 00 v1 + e(A (t) + X) v2 + X v2} e 1"
and
T
Re{Me, v9 v) = Re j (ax{t\ X v1 - v2, v1) + [A (t) v1 + e(A (t) + X) v1 +
o
T
+ Xv2,v2])dt = X \ {ait'.vi.v1) + X[v*]2 - Re[v\v2] + [v2]2} dt +
o
T
+ Re j {-a{t;v2t v1) + [A{t) v\v2]} dt +
o
T
+ e [ ax(t; v2, v2) dt
o
T
= X j {ax{t; v\ v1) - Re[v\ v2] + [v2]2} dt +
o
T T
+ e j ax(t; v2} v2) dt ^ X j {<x \\ v1 \\2 + $[v2]2) dt +
0 0
T
+ eotj\\v2\\2dt.
o
Therefore
(8.39) Re{M,v,v) ^ <*x \\v\\%> + eoc j \\v2\\2dt ^ min(alf ea) \\€\\*..
o
Therefore hypothesis (1.9) is satisfied, thanks to the introduction
of the regularizing term. □
The semi-group G(s) and the operator A.
For veJti? (for example), we define:
v(t - 5) e-"s, t > s
(8.40) G{s)v =
te]0,T[.
0, t < s
The infinitesimal generator (formally) of G(s) is of course given by
(8.30), its domain consisting of the functions which vanish at the origin.
The fundamental point is to see that G(s) is a contraction semi-group

8.3 Remarks on the Application of the General Theory of Section 1 273
in J? (the same proof will yield the result in y and in y), for the norm
corresponding to (8.36). That is, we have to verify that
T
l|G(s)ii£, = jaA(t''vl{t - 5)>^(* - s))e~2*sdt +
s
T
+ f \v2(t - s)\2e-2>sdt ^
s
T
g> j ak(t;v1{t),v1[t))dt+ j \v2{t)\2 dt.
o o
Of course, it suffices to show that
T T
e~2/is f *xQ'>vl(t - s),*1^ - s))dt ^ \ ak(t\v1{t)tv1{t))dt
s 0
or (eliminating the index " 1") that
T-s
(8.41) 0{s) = e-2»s j ax(t + s;v(t),v(t))dt ^ 0(0).
o
But
r T-s
0'(s) = e-2"s
T-s
-2/j, f ax(t + s; v(t), v{t))dt +
- e-2>sax(T',v{T -s),v{T- s))
+ j a'A(t + s;v{t),v{t))dt\
o J
and, thanks to the third condition in (8.29), we have
(8.42) 0'(s) ^0.
From which (8.41) and the desired result follow. □
Remark 8.3. Inequality (8.41) holds if
a(t + s; v, v) ^ a(t;v, v), i.e. if
(8.43) the function t -► a{t\v,v) is decreasing, Vv e V.
If (8.43) holds, then, with no differentiability hypothesis on a(t; v, v),
we have existence and uniqueness of a solution for problem (8.34). □
We may now apply Section 1; we obtain
Proposition 8.1. There exists a unique solution UgE^ n D(A', Y')
of (8.39) for {0,g}ef. D

274 8. Equations of the Second Order in t
Furthermore, if g e L2(0,T; H), we deduce from (8.39) that
'* "f} verifies, as e -» 0:
in a bounded set of Z,2(0, T; V),
(8.44) \ u] remains
[ u2t remains in a bounded set of L2(0, T\ H).
Therefore, we may extract from ue a sequence, still denoted by
ue, such that ue -» w weakly in 3f. But we can specifiy (8.34):
(8.45) A uc + MUe + e{0, (A(t) + A) u2e} = {0, g}.
We introduce
L2(0, T;D(A{t))) = {v | v eL2{0, T; V)t v(t) eD(A{t)) = domain of
A® in H a.e., A{t) v{t) eL2{0,T; H)}.
Then
(A(t) + X)ul->(A{t) + A) ze;2 weakly in (Z,2(0, T; Z)(il*)))'
and therefore
e{Q,{A(t) + X)u2e}^0
in, for example, the space
L2(0,T;V)x(L2(0,T;D(A(t)))f
and (8.45) yields
(8.46) A& + Mw = {0,g}, ze>e.?f
in the limit.
If w = {w1, z#2} and if we set w = w1, w is a solution of
^2z£; ^z£;
+ ^4 (t) w + 2& + k2 w = g
dt2 dt
and satisfies w(0) = 0 (for u'e(0) = 0 and ue(0) -» ze>(0) weakly in #).
Finally, we verify that w' (0) = 0 by passage to relations "integrated
in t'\ as at the end of the proof of Theorem 8.1. □
Remark 8.4. The preceding regularization could be called "parabolic
regularization".
Of course, we may then apply elliptic regularization to equation (8.34),
which yields
(8.47) Aue + Meue + e1A*Aue = {0,g}, e9et > 0. Q

8.4 Additional Regularity Results
275
8.4 Additional Regularity Results
Our aim is to show the following result:
Theorem 8.2. Assume that (8.1) and (8.2) hold. Then, after possibly
a modification on a set of measure zero, the solution u of (8.3), (8.4), (8.7),
(8.8) satisfies
(8.48) u = ji^l eC°([0, T]; V) x C°([0, T]\ H),
{/, u0, Ui) -► u being a continuous mapping of
L2{0, T',H)xVxH-+ C°([0, T]; V) x C°([0, T]; H).
Remark 8.5. Other regularity theorems, for the "concrete'' case of
differential operators, will be found in Chapters 4 and 5 of Volume 2
(and results of Mk regularity,'' abstract'' and'' concrete'', in Volume 3). □
We shall first prove some lemmas.
Lemma 8.1. Let X, Y be two Banach spaces, X a Y with continuous
injection, X being reflexive. Set:
Cs(0, T;Y) = space of functions / e Z,°°(0, T; Y) which are scalarly
continuous1 mappings of [0, T] -► Y.
Then
(8.49) L°°(0, T\ X) n Cs(0, T; Y) = Cs(0, T\ X).
(Therefore if feL»{0, T; X) n CS{0, T; Y),f{t) - which is defined
in Y for all t — is, in fact, in X and / is scalarly continuous and takes
its values in X.)
Proof. 1) We show that f{t) eX and that
(8.50) ll/Wllx ^ constant.
We may always assume that / is defined on Rt, with properties
analogous to those of / on [0, T].
Let q„ be a regularizing sequence of even functions of 2 (Rr), with
jQn(t)dt= I.
Rt
Since / e Z,00 (Rf; X), f * qn satisfies
II / * fti(0 II x ^ constant = M.
Therefore, for arbitrary fixed t we can find a sequence v such that
(X being reflexive):
/ * (?v(0 "* /(0 weakly in X
1 That is *-» </W,/> is continuous on [0, T], V/ € r, dual of Y.

276 8. Equations of the Second Order in /
and
(8.51) \\f(t)h^M.
But, for arbitrary y'eY', t -+ (f(t),y'} is continuous, therefore
</* Qn(t) -f(t), y'> = (a, * </.y'» (t) - </, y'> (<) - 0
and so, in particular
/*£,$ -+/W weakly in Y.
Therefore /(*) = f(t) and (8.51) yields (8.50) with the constant equal
to M.
2) Now we show that / is a scalarly continuous function of [0, T] -► X.
If tn-+ t, we can (since ||/(OlU = ^0 extract tv such that f(tv) -► %
weakly in X. But f(tv) -► f(t) weakly in Y, therefore % = f(t) and the
lemma is proved. □
Lemma 8.2. // u satisfies (8.3), (8.4), (8.7), (8.8) we can always assume
[after possibly a modification on a set of measure zero) that
(8.52) **eCs(0,r;F), -^eCs(0,T; H).
at
Proof. It is known that «el°°(0, T;7) and u' eZ,°°(0, T\ H),
therefore, in particular, u is (after possibly a modification on a set of
measure zero) a continuous mapping of [0, T] -* H, therefore ue
e Cs(0, T\ V) by Lemma 8.1 with X = 7 and Y = tf.
Then, u" = f — A u eL2(0, T\ V), therefore w' is a continuous
mapping of [0, T] -+ 7', whence u' e Cs(0,T; H) by another
application of Lemma 8.1. □
Lemma 8.3. (Energy equality). Let u verify (8.3), (8.4), (8.7), (8.8)
and (8.52). Then, for all t:
(8.53)
a(t;u(t),u(t)) + \u'(t)\2 = a{0;uo,uo) + |%|2 +
t t
+ \a'{o\ u, u) da + 2Re \(f,u')da.
Proof. - 1) Notation. We take t = *0 in (8.53).
We introduce 0 = <9d = 0j(*) = {1 in [d, t0 - g], 0 outside [0,*0],
linear on [0, d] and [t0 — d, t0], continuous on Rj; q = Qn =
regularizing sequence of even functions, \ Qn(t) dt = 1.
We shall assume that A (t) is defined for all t e R — with the same
properties as on [0, T].

8.4 Additional Regularity Results 277
In the same way, we shall assume that u is defined on Rt, with
properties analogous to those of u on [0, T] (which is permissible by
"extension by reflexion").
We recall that [, ] denotes the scalar product in H or the anti-
duality 7', 7; we shall denote ( , ) the antiduality L2 (Rt; V), L2 (Rt; 7)
or the scalar product in L2(Rt',H).
2) We have
' (A' (t) {q * (0O u)), Q * (0O «)) + 2Re{Q * (0O Au),q* (0o u')) +
+ 2Rq(A(q * (0O u)) - q*(A00u),q'* (0o «)) +
+ 2Re[(Q* q* (®0 A u)) (0),u(0)-] -
[ - 2Re[{Q *q(0oA u)) {t0), "{to)] = 0.
Indeed, starting with
+ 00
d
(8.54)
i
rf/
— oo
we obtain
2Re(A(Q*d)u)t (q*0u)') + (A' {q* 0 u), q* 0 u) = 0
and thus
[ (4'fe*^),^^) + 2Re(e* {A(9u),q* (0«')) +
(8.55) j + 2Re(e* (4 0^),^* (0'«)) +
( + 2Re(^(^* (0«)) - g* (4 0«),g'* (0«)) = 0.
Now we let d -► 0 in (8.55).
In the first, second and fourth terms of (8.55), it suffices to replace 0
with 0O. The third term may be written
(8.56) 2Re(e*(i4(0-0o)«),e*0'«) + 2Re(e*(i4 0o«),e*0'«).
But (0 - 0O) « -► 0 in L2 (Rt; 7), therefore £ * (A (0 - 0O) «) -► 0
in L°° (Rt; 7') (the supports are compact) and since q * (0' u) is bounded
in Z/(Rt; 7) /since J" |0'| rf* = 2V the first term of (8.56) goes to zero.
The second term is equal to
2Re(e*£* [A (90u),(9' u).
But t -> [q * q * (A 0O u) (t), u (t)] is continuous, so that this
expression tends towards
2Re[(e*e*(0o^«))(o),«(o)]-2Re[(e*e*(0o^«)(O»«W)].
Whence (8.54).

278 8. Equations of the Second Order in t
3) We have
\ 2Re{Q*{V0u")>Q*{&0u,)) +
(8.57) J + 2Re[(e * q * (0O u')) (0), u' (0)] -
[ - 2Re[(e * q *(0O *')) (*0), «'(*<,)] = 0.
The idea of the proof is the same as for (8.54). We start with
+ 00
i
(q* (&u'),q* {6m'))dt = 0
(8.58)
from which we obtain
2Re(e * (0 u'), Q * (0 u")) + 2Re(e * (0 - (9Q u') ,q * 0' u') +
+ 2Re(g*(0oiO, e*(0'iO) = 0.
As (5 -► 0, the second term of (8.58) -► 0, the first tends towards
2Re(e* {<9Qu')tQ* (0Qu")).
The last term is equal to 2 Re (q * q (ff0 u'), & u') and tends towards
2Re[e * e * (0O u') (0), «'(0)] - 2Re[^ * g * 0O «'(g, «'(*0)],
since / -► [g * p * @0 u' (t), u' (t)] is continuous. Whence (8.57).
4) We add (8.54) to (8.57); taking account of the fact that A(t)u + u"
= /, we obtain:
f (A'{q* (0O *)), Q * (^o «)) + 2Re(g * (0O /), g * (0O *')) +
+ 2Re((^4 (g * (<P0 u)) - q * (^4 0O «))', p * 0O u) +
+ 2Refc * £ * (0O -4 u) (0), «(0)] -
- 2Re[q * ^ * (0O 4 «) (*0), u(toy] +
+ 2Re[Q*Q*{00W) (0),«'(0)] -
-2Re[Q*Q*{0ou') {t0),u'{t0)] =0.
(8.59)
According to the "vector" Friedrichs' Lemma (see Lions [13],
Lemma 7.2, p. 72, for example), the third term of (8.59) tends towards 0.
The first and second terms tend towards {A'G0)GQu)+2 Re (0O /, 00 u')
and therefore then remains only to pass to the limit in the four last
terms. Since 0 and t0 play symmetrical roles, we shall have (8.53) if
we can verify that (setting q * q = a):
(8.60) 2Re[(r* {<D0.A u) (t0),u(t0)] -> a(t0;u{t0), u(t0)),
(8.61) 2Re[a*{G0u,){t0),u'(t0)] ->|«'(*0)|2.

8.4 Additional Regularity Results 279
But (a = an) :
to
2Re[an * (0O A u) (t0),u(t0)] = 2Re j an(t) [A u(t0 - t),u(t0)] dt
o
and since an is even,
to
joH(t)dt = i,
o
therefore
2 Re [an * (0O 4 «) (*0), u (t0)] - |>4 (*0) u (t0), m (*0)]
t
= 2Re J (T„(/) [(4 «) (*0 - 0 - (A u) (t0),u(t0)] dt,
o
which goes to 0 as dn -► d, whence (8.60) — and analogously for (8.61).
This ends the proof of Lemma 8.3. □
Proof of Theorem 8.2. We already know that (8.52) holds.
On the other hand, according to (8.53), the function
t-+a(t;u(f), u(f))+ \u'(f)\2
is continuous on [0, T] and since t -► u (t) is a strongly continuous
function of [0, T] -► H, we see that
(8.62) t-+ cp(t) = ak(t\ u(f), u(t)) + \u'(t)\2 is continuous on [0, T\.
Let tn -► t and
£. = \«'{Q - **'(0I2 + *x{tn\ <Q - <t), HQ - u(tj).
We have
fi. = ^(0 + <p{t) + 2Re[^(/„; u(t),u(t)) - ak(t\ u(t),u)J] -
- 2Re[u'(tn), u'{t)] - 2Re^(/; w(0, ^W) -
- 2Re [ax(tn; u{tn),u(t)) - ^(/; u{tn),u(t))\.
But
|^(/„; «(/), «(*)) - ax(t\ u(t)fu(t))\ <C\t-tn\}
\ax(tn; u(tn),u(t)) - ax(t\ u(tn),u(t))\ £C\t-tH\,
and therefore
5n - 2<p(t) - 2Re(|^(/)|2 + ax(t, u(t),u(t))) = 0.
Since fB ^ \u'(tn) - w'(/)|2 + * \\u(tn) - w(/)||2, we have the
theorem. □

280
8. Equations of the Second Order in /
8.5 Parabolic Regularization; Direct Method and Application
We take up problem (8.3), (8.4) again and associate to it, as in
Section 8.3, the regularized parabolic problem
(8.63) A (t) ue + < + e(A (t) + X) u'B = /, e > 0,
(8.64) uB{0) = 0, 4(0) =%.
Theorem 8.3. Assume that (8.1), (8.2), (8.5), (8.6) hold.
1) Then, for every e > 0, problem (8.63), (8.64) admits a unique
solution ue which satisfies
(8.65) uteC°([0,T];V),
(8.66) <el2(0, T\ V) n C°([0, J]; H).
2) As e-^0,
ue^> u uniformly in C° ([0, T]; F),
^ -► w' uniformly in C° ([0, T]\ H),
(8.67)
w/^re ^ zs *A* solution of (8.3), (8.4), (8.7), (8.8) (solution which belongs
to C°([0, T]; V) and has derivatives in C°([0, r];#), according to
Theorem 8.2).
Proof of Theorem 8.3, first part. As in Section 8.2, we can use the
method of Faedo-Galerkin. In the notation of Section 8.2, we introduce
m
«««.(*) = E&mP) »i. satisfying
(8.68) «(*; «„,, »j) + [<;. wj + e ^(f; u'em, »,) = [/(*), wj,
1 ^ j ^ m;
(ax{t; u, v) = a(t; u, v) + X[u, v\), with gim(0), g'tm(0)
as in (8.18), (8.19).
Multiplying (8.68) by g'im(t) and summing over /, we obtain:
d
(8.69)
rf*
(a(t;utm(t),uem(t)) + \u'sm(t)\2) + 2sax(t;u'em,u'sm) -
- a'{f. *m. um) = 2Re[(/W. u'em(<))],
from which we deduce (as in Section 8.2, first part of the existence
proof) that
as m -> oo, wEOT (resp. ^m) remains in a bounded set of
L°°(0, T; V) (resp. L2(0, T; F) n L°°(0, T\ H)).
(8.70)

8.5 Parabolic Regularization; Direct Method and Application 281
We then extract uefl such that
dum
du0
dt
dt
uefl -► ue weak star in U* (0, T; V)
weak star in Z,00 (0, T\ H) and L2 (0, T\ V).
We verify that ue is a solution of (8.63), (8.64).
CLM
We already know that ue eZ,°°(0, T\ V), — eL2{0, T; V),
dt
therefore ue is (after possibly a modification on a set of measure zero)
a continuous function of [0, T] -» F. Now
*;' = / - A (t) uE - e(A (t) + X) u'6
and since «e'el2(0, T; F), we have
(8.71) u'e'eL2(0,T;V),
which, together with ueeL2(0, T; V) and Theorem 3.1 of Chapter 1,
shows that (after possibly a modification on a set of measure zero)
ue is a continuous function of [0, T] -» H. This shows the existence of
a solution we = we having the stated properties. □
Let us verify the uniqueness. In fact, we obtain from (8.63) that
— (*(*; uE(t)}uB(t)) + \u'e(t)\2) - a'(t\ uE(t),u£(t)) +
whence
(8.72)
+ 2e ax(t) u'e{t),u'e(t)) = 2Re[/(t), u'e(*)] a.e.,
r
«(*;«.(*).««(*)) + KWI2 - J«'(^;«.(e)»«.(e))^ +
0
r
+ 2e f «A((r; «i((r), «i(o,))io' = a{0; u0, u0) + |%|2 +
o
t
+ 2Re j[f{a),u'g{a)]da,
this last equality in fact holding everywhere, thanks to the continuity
properties of ue and ue.
Uniqueness follows immediately from (8.72): if u0 = 0, uY = 0,
/ = 0, it follows that ue = 0. □
Proof of Theorem 8.3, second part. It follows from (8.72) that
uE (resp. u'e) remains in a bounded set of £°°(0, T\ V)
(8.73)
(resp. L°°(0,r;H)) as e -► 0

282 8. Equations of the Second Order in t
and that
(8.74) \ eu'e remains in a bounded set of L2(0, T; V).
Therefore, we can extract a sequence — still denoted ue — such that
ue -> u weak star in Z,00 (0, T\ V)
K ~+ W weak star in Z,00 (0,T; H).
Then
u" = f-A (t) ue - e(A 00 + A)< -+f-A(t)u weakly in L2 (0, T\ V)
(for example), therefore («)" + A (t) u = / and
ue(0) -* «(0) weakly in iZ (for example)
and
u'e(0) -* («)' (0) weakly in F' (for example),
so that «(0) = w0> W (0) = «i, therefore w = u and finally we have,
without extracting a subsequence:
Iue -+ u (resp. w' -► w') weak star in Z,00 (0, T; V)
(resp. z£>£#& star in Z,00 (0, T; iZ)).
There remains to show the uniform convergence.
We introduce
(8.76) cpE = uB- u\
we have
(o./7) {
ly.ei-tO.TjF), <p'eeL°>{0,T;H), %(0)=0, <p'.(0) = 0.
According to the energy equality (Lemma 8.3):
t t
a(t'> 9>e(0, ?>«(')) + l^iWI2 = J «>; <Pe> %) d° - 8\ al(a'> Ue> V,) d(T>
0 0
whence
t I r I
II P. W II2 + ItfWI2 ^Ctj \\<pe(o) ||2 rf<r + Cx e J* «A(a; u'„ <pe) del.
o |o I
By GronwalTs inequality, it follows that
(8.78) Up.M||a + \<p'e(t) |2 £(exp(C*))Clfi
j <*x(<*m>K >%)*<*
I i \ 1/2
^ (according to (8.74)) C2 e1/2 J || <pe(t) \\2 dt\ ^C3e
1/2
1/2
from which (8.67) follows. □

9.2 Transposition 283
9. Equations of the Second Order in t; Transposition
9.1 Adjoint Isomorphism
As usual (see Section 2.2), we first consider the adjoint isomorphism.
For cp given in L2(0, T\ H), let v be the solution of
f Alt) v + v" = <p,
(9.1)
I v{T) = v'{T) = 0.
We define:
(9.2)
(9.3)
X = space described by the solution v of (9.1) as <p describes
L2(0,T;H).
According to Theorems 8.1 and 8.2, the functions of X have the
following properties:
veC°{[0,T];V), v'eC°{[0,T]',H),
v" eL2{0,T;V).
Providing X with the topology carried over by the mapping <p -+ v,
we have (done the necessary to have) the result:
(9.4) A (t) H is an isomorphism of X onto L2(0,T; H).
dt
9.2 Transposition
From (9.4) we deduce
Theorem 9.1. Hypotheses of Theorem 8.1. Let L be a continuous
antilinear form on X. There exists a unique u eL2(0, T\H) such that
(9.5) (u,A(t)v + v") = {L,v) VveX,
(where the first parentheses denote the antiduality between L2 (0, T, H)
and itself and the second between X' and X). D
Now we have to choose L.
Formally (compare with (4.5); see also (3.11), Chapter 6, Volume 2)
we shall take
T
(9.6) (L,v) = J [/(t), v(t)] dt + [ultv(0)] - [«o- »'(0)].
0
giving suitable meanings to the various scalar products occuring in this
formula. Still formally, we then see that the solution u of (9.5) satisfies
(9.7) A{t)u + u" = f
(9.8) «(0) = u0, u'(0) = ux.
Now we have to make this rigorous.

284 9. Equations of the Second Order in t; Transposition
9.3 Choice of L
Compare with Section 4.5 and with the introduction of 3 in (4.24).
Let q be the function defined by (4.23). We define the space 3 by
f 3 = {v | v e L2{0, T\ V), q V eL2{0, T; H),
(9.9)
| Q2v"eL2{09T;V'),v{T) =v'{T) = 0}
[provided with norm
T \ 1/2
j(\\v(t)\\2+Q2\v'(t)\2 + Q4\\vf'(t)\\2v,)dt\
which makes it a Hilbert space) . We have:
Proposition 9.1. The space £&(]0, T[; V) is dense in 3 {defined by
(9.9)).
Proof. Let Gn be a sequence of scalar functions belonging to C2 ([0, T]),
0n vanishing in (0,e„),
en-+0 if n-^oo, 0n = \ if t ^ 2en with \q<P'„\+ \Q2(9'n'\ ^ C.
Then if v e3, (9n v -► v in 3 (since, as is easily seen,
Q<9'nv-+0 in L2{0,T;H),
q2&; v + 2Q2(9'nv' -.0 in L2{0, T\ V')).
Since v(T) =0, v'(T) =0, we may truncate in the neighborhood
of T; finally, by regularization in t, we obtain the desired result. □
Corollary 9.1. The space 3' (anditual of 3) is a space of distributions
on [0, T] taking their values in V. Every f e3' may be represented, non-
uniquely, by
(9-10) f = f0+J-{Qfl)+JL(Q2f2)>
where
/06L2(0,r;F), /iei:2(0,r;i/), f2eL2(0,T;V). D
Since, still according to (9.3), we have
(9.11) XczS,
we see that
{for f e3', v -> (f,v) (scalar product between 3' and 3)
is a continuous antilinear form on X. □

9.4 Trace Theorem 285
Also, since v e C°([0, T]\ V) and v' e C°([0, T]\H), we may take
(9.13) %eF, w0e#
in (9.6).
Summing up, we have:
Proposition 9.2. In (9.6), w way tafo feS', u0eH and u± e V.
In particular, we may take f = /0 e L2(0,T] V). U
Now the problem is to interpret equation (9.5) for the preceding
choice of L, which we shall do (in opposition to what we have done in
the parabolic case, see Section 4.6), for a particular choice of /
(/ eL2(0, T; V')). The general case presents some difficulties (see
Problem 13.11).
9.4 Trace Theorem
We shall make an additional hypothesis:
I the domain D(A (t)) of A (t) in H is independent of t and, VveH,
t -» A (t) A (0) l v is a continuous function of [0, T] -► H.
Let us be somewhat more precise; D{A (t)) in H is the set of u's e V
such that A(t) u e H] it is assumed to be provided with the norm of
the graph:
(\u\2 + \A(t)u\2y* = \\u\\D(«,»;
then D(A(t)) = D(A(0)) with norm equivalence, and this uniformly
in t, therefore
C~l IMIduco)) ^ NIIdu(d) ^ C \\u\\D(M0»
\/ueD(A(0))t te[0tT]. Q
Remark 9.1. In the applications (see, among others, Chapters 4 and 5
of Volume 2), hypothesis (9.14) is satisfied when A(t) is an elliptic
operator to which one associates the Dirichlet boundary conditions. □
Under hypothesis (9.14), we have
(9.15) A(t)eSe{D{A(Q))\H)
and by transposition, since A(t) = A*(i):
(9.16) A(t)eX(H;D(A{0))').
Furthermore, \\A (t) ||^(du(0));H) ^ constant and therefore the same
is true for \\A(t)\\^{H.DiAi0)y). Also, D(A (0)) is dense in V, therefore:
(9.17) D(A(0)) cVcHcV cD(A{0))'.
Lemma 9.1. // u is given in L2(0,T, H), we have:
A(t)ueL2(0,T;D(A(0)y).

286 9. Equations of the Second Order in t; Transposition
Proof. According to the preceding remarks, we only need to show
that t -+ A(t) u(t) is a measurable function of [0, T] -► D(A (0))'. Since
we have assumed (see Section 4.2) that V is separable, H is separable
and, A (0) being an isomorphism of D(A (0)) onto H, D(A (0)) is
separable. Therefore, it is sufficient to show that A (t) u (t) is scalarly
measurable, i.e. that for arbitrary v e D(A(0)), the function
t-+(A(t)u(t),v)
is measurable. But it is equal to (u (t), A (t) v) and it suffices to verify
that t -+ A(t) v is measurable with values in H.
But t -> A (t) v = A (t) A (0)" l(A (0) v) is continuous in H, according
to (9.14), whence the result. □
Remark 9.2. If D(A(t)) depends on t, we have A (t) u e
e L2(0,T; D(A(t))'), if we interpret this last space in a suitable
way. D
We then have the following (partial) result:
Theorem 9.2. Assume that hypotheses (8.1), (8.2) are satisfied and
that (9.14) holds. Then, if f is given with
(9.18) f = f0eL2(0,T;V)
and u0 and ul are given with (9.13), the solution u of (9.5) satisfies
(9.7a) A(t)u + u" = f in 9'Q0,T[;D(A(0))')
and
u(t) -+u(0) = u0 in [H, F']1/2 as t-> 0
«'(*)->«'(()) =«! in [y'tD(A(0))%2 as t-+0.
Proof. 1) In (9.5), we take v given by
v{t)=<p{t)k, <pe@(]0,T[), keD(A (0)) (therefore veX).
(9.7a) follows. Then, applying Lemma 9.1, we obtain:
(9.19) u" =/ - A{t)ueL\0,T}D(A(0)y)}
from which, by the intermediate derivatives theorem of Chapter 1,
Section 2 and since [H, D(A (0))']1/2 = V (Chapter 1, Sections 2.4
and 6), it follows that
(9.20) u' =L2(0,T;P)
and therefore (Chapter 1, Section 3) u is a continuous function of
[0, T] -► [H, V']l/2 and u' is a continuous function of [0, T] ->
^[V',D{A{0))']ul.
(9.8 a)

9.5 Variant; Direct Method 287
2) Now if we take v defined by
v(t) = V(t)k, <pe9(t09 T])$ <p(T) = <p'(T) = 0, keD(A(0))t
(then v e X),
it follows from (9.7 a) that
T
j {A (t) u + u", v(*)> dt = (/, v) = (u, A(t)v + v") -
-<«'(0),"^0)> + <u(0)tVJO)}.
But according to (9.5):
(u, A(t)v + v") = (/, v) + <%, ^0)> - (u0t 7(0)>
and therefore
<«! - «'(0),*>y(0y- <w0 - «(0),*>y'(0) = 0,
V9? e 0([O, T]) (with y (r) = <p' (T) = 0) and V* e D(A (0)),
whence (9.8 a). D
Remark 9.3. The result is partial in the sense that we ignore whether
without additional hypotheses on A (t), u is the unique element of
L2(0, r; H) to satisfy (9.7a) and (9.8a) (see Section 9.5 on this point).
9:5 Variant; Direct Method
We start with a remark of a general nature:
Remark 9.4. Transposition — which we use systematically in this
book — is only one procedure for carrying out certain passages to the
limit. It maybe of interest to carry out these passages to the limit directly.
This becomes particularly useful when the ''natural spaces'' of
regularity are not reflexive — which is precisely the case for the operators
d2
A(i) H (since in this case we meet the spaces L°°(0, T\ V) or
C° ([0, T]; V)). In this section, we shall give an example of direct passage
to the limit. □
We shall assume that (8.1) and (8.2) ,with X = 0, are satisfied (in
order to simplify the notation, but this is in no way essential) (see
Sections 8.3 and 8.5).
We want to solve, in a suitable way, the problem (see Theorem 9.2):
A (t) u + u" = /
u(0) = u0, u'(0) = ult
with
(9.22) /el2(0, T\V)9 u0eH, «l67'.
(9.21)

288 9. Equations of the Second Order in t; Transposition
Theorem 9.3. Assume that (8.1), (8.2) (with A = 0) are satisfied and
that /, u0> ul are given with (9.22). Then, there exists a u satisfying (9.21)
and
(9.23) weL°°(0,r;^),
(9.24) *'eL"(0,r;7').
Furthermore (see Remark 9.10) we shall see that
ueC°([0,TytH), u'eC°([0,T];V'). D
Remark 9.5. The function ^4 (t) u belongs to the dual of
( L2(0, T;Z)(i4(*))) = {v | veL2(0, T; V), v(t) eD(A (t)) a.e.,
j A(t)veL2(0,T;H)};
it is in this sense that we can interpret (9.21). We can also replace
(9.21) with
(9.21a)
j [u, A(t)v + v"] dt = j [/, v] dt + [ultv(0)] - [u0, v'(0)]
[ Vv e X (defined in (9.2)) and such that v e L2(0, T; D(A (t))).
(In (9.21a), [f,v] denotes the function t -► [f(t), v(t)], the brackets
then denoting the antiduality between V and V; the second (resp.
third) brackets express the antiduality between V and V (resp. H
and itself).) □
Remark 9.6. The problem of the uniqueness of the solution of (9.21 a)
is solved only under an additional hypothesis on A (t) (see
Theorem 9.4). □
Proof of Theorem 9.3. — 1) We consider (which is permissible):
(9.26) /fl€l2(0,T;i/), u0neV, ulneH,
with
(9.27) {/.,«o.,«i.}^{/o.«o.«i} in L2(0, T; V) x H x 7'.
We then consider the problem
f A W w» + ^ = '■
(9.28)
I «»(0) = ^on, <(0) = «iB.
Thanks to (9.26), we have (Section 8):
(9.29) un e L00 (0, T\ V), < e L00 (0, T; H).

9.5 Variant; Direct Method
289
2) A priori estimates. — We take the scalar product of the two terms
of (9.28) with A(t)-lu'n; according to (9.28), u'n(t)eV a.e. and
A (t)_1 u'„(t) e V a.e. (in fact everywhere if we apply Theorem 8.2). Then,
since
[A (t) un (t) ,A{t)-lu'n (t)] = [un (t), u'n (t)],
we obtain
1 d
'lit
\un(t)\2 + Re WW. A(t)-> u'M = RtlUQ.Ait)-1 <(t)],
or, again a.e.
2 at
-^(A(t)-i)un(t),un(t)
= Re(fn(t),A(t)-1u'n(t)]
whence
(9.30) \un(t)\2 + \A (t)-1 <(t),<(t)] = K„|2 + [A (0)"1 uln, uln] +
da + 2Re
[fn(<y),A(o)-iu'n(<y)]da.
But
[A{t)~l v, v] ^ q \\v\\l, q > 0, where \\v\\+ = norm in V',
and
da
Aia)-1 = -Afa)-1 A'(a) A(a)-X eX{V']V)
and remains in a bounded set of J?(V; V), therefore, in particular, also
in a bounded set of &(H\H).
Therefore, we deduce from (9.30) that
(9.31) KMP + KWIli^
KJ2 + ll«i„lli + J>»l2^
+ CX
/[ilW-M'^ilW-V.W.^W]^
It follows that
(9.32) \un(t)\2 + \\K(t)\\l^C2
KJ2 + II«i.llj +J" II/„ml do

290 9. Equations of the Second Order in t; Transposition
Remark 9.7. In fact, we obtain somewhat more. In (9.32), we may
replace
T
j\\U<y)\\Ua with ||/.n
0
Remark 9.8. Inequality (9.32) is in a way a "shifted energy
inequality", taking place in spaces in duality with V and H. This procedure
is classical for the Cauchy problem for hyperbolic operators, Leray [1]. D
3) From (9.27) — (9.32) we deduce that un (resp. u'n) remains in a
bounded set of L°°(0, T\ H) (resp. L°°(0, T\ V) as n -► oo. Therefore,
we may assume, by extracting a sequence uM, that
u^-* u weak star in Z,00 (0, T\ H),
u'r -+ (*)' weak star in Z,00 (0, T\ V).
Let us take v as in (9.21a). Then, we deduce from (9.28) that
T T
(9.34) J [«,, A(t)v + v"] dt = J* [/„, »] dt + [ulni v(0)] - [u0n, »'(0)]
0 0
and passing to the limit in (9.34) (for w = //), we obtain that u satisfies
(9.21a), (9.23), (9.24).
Therefore we may take u = u and we have the theorem. D
Remark 9.9. Following Remark 9.7, we see that Theorem 9.3 still
holds (in the interpretation (9.21a)) under the hypothesis
(9.35) fe(L2(0,T;D(A(t))))' [see (9.25)]. D
Remark 9.10. Inequality (9.32) gives a little more than (9.23), (9.24):
as n -► oo, un (resp. u'n) converges uniformly to u (resp. u') in H (resp. V)
and therefore:
under the hypotheses of Theorem 9.3, u e C°([0, T]\ H)
and u' eC°([0,T]; V). D
Remark 9.11. Of course, {/, u0> ux) -* {u, u'} is a continuous
mapping of
L2(0, T; V')xHxV'-+ I°°(0, T; ff) x L°°(0, T\ V). D
Theorem 9.4. Assume that (8.1), (8.2), (9.22) hold. Furthermore,
assume that
(9.37) Vu,v eV, the function t -+ a(t]u,v) is in C2[0, T],
Then problem (9.21) (or (9.21a)), (9.23), (9.24) admits a unique
solution.
(9.33)
(9.36)

9.5 Variant; Direct Method 291
Proof. It is sufficient to show that (9.21a) holds for all v e X, that
is that the subspace
X0 = {v\veX,ve L2(0, T] D(A (t)))}
is dense in X (for the terms of (9.21a) are continuous in the topology
of X). Now, let veX] set
g = A{t)v + v", geL2(0,T;H),
and consider gne C1 ([0,T]] H) (for example), gn(T) = 0, with
gn-+g in L2(0,T,H).
Consider v solution of
Then DBeI and vn -+ v in X. But we can differentiate (9.38) m £;
it follows that
v'neL2(0,T;V), ^el2(0,r;i7).
Theni4(/) v„ eL2(0, T;i7), vB(/) eZ)(4(*))a.e. and therefore vneX0>
whence the theorem. D
Remark 9.12. We can regularize the data and the operator
simultaneously and consider, instead of (9.28), the regularized parabolic
operator (see Section 8):
(9.39)
*{*)"» +< + **{*)< = L
^£n(0) = u0n, <„(0) = uln.
It can be shown that uBn (resp. ufen) converges uniformly in
[0, T]-+ H (resp. V) as n -+ oo, e -+ 0. D
Remark 9.13. Of course, the generalization of Theorems 9.1—9.3
is not the f<ultimate generalization"] for example, we could take the
scalar product of the two terms of (9.28) with A (t)~2 u'n, A (t)~3 u'n>...
to obtain inequalities of the type (9.32), but where un(t) e V, un{t) e
eD(A(t)y, etc., which allows passage to the limit under weaker
hypotheses on /, u0, ux (we shall go even much further in the case of
differential operators — see Volumes 2 and 3). □
Remark 9.14. The preceding method also applies to the equations
of the first order of Section 4, under hypothesis (8.2). □
Remark 9.15. We may also interpolate between the results of Sections 8
and 9.

292 9. Equations of the Second Order in t\ Transposition
According to Theorems 8.2 and 9.3 (with (9.36)) respectively,
{Awo'Ki} -* {u,W}
is a continuous mapping of
L2(0, T;H)xVxH-+ C°([0, T]; V) x C°([0, T], H)
and of
L2(0, T; F) x H x 7' -* C°([0, rj; tf) x C°([0, T]; F).
By interpolation and using (see Chapter 1, Section 14.2):
[L2(0, T; ^),L2(0, T; V% = L2(0, T; 7~')
(where
[F,tf]a = 71-, (F1"")^ ^-x)
and
[C°([0, T]; F), C°([0, T]; #)]* = C°([0, T]; F1"*),
we obtain
Theorem 9.5. Assume that (8.1), (8.2), (9.37) hold. f, u0, ul are given
with
(9.40) /eL2(0,r;F-*), O<0 < 1,
(9.41) u0 e Vl~\ u, e V-d.
Then, the solution u of (9.21a), (9.23), (9.24) satisfies
(9.42) ueC°{[0,T];V1-9),
(9.43) u' eC°([0, T]\V-9). D
9.6 Examples
We shall give some examples of applications of the preceding
theorems. As we have done for the equations of the first order in Section 4.7,
we could apply either the regularity theorems of Section 8, or the more
general theorems of Section 9, by transposition (Theorems 9.1 and 9.2)
or directly (Theorems 9.3 and 9.4).
We shall limit ourselves to giving the results obtained through
application of the theorems of Section 9.
9.6.1 Example 1
Let Q be a bounded open set in Rn. In this example, Q is arbitrary,
therefore the boundary r may be "arbitrarily irregular".
We set
du dv
(9.44) a(t;u,v) = £
■dx,
dxt dxt

9.6 Examples 293
for
u.veV = Hl(Q).
If H = L2(Q), then V = H~l(Q).
Let /, u0, «! be given with
(9.45) feL2(0,T;H-i(Q)),
(9.46) u0eL2{Q), UleH-l{Q).
Then we know that there exists a unique ueL2 (Q) satisfying (see
Theorem 9.1):
T
(9.47) «(—VT-Av\dxdt= [/(<),»(<)] ii +
+ [«i(0),»(0)]-L(0).-^-(0)
where [, ] denotes the antiduality between H~1 (Q) and Hq (Q) (and
the scalar product in L2(Q)), for every function v eL2(0, T; Hq(Q))
such that
^-eL2(0,T;L2(Q)) = L2(Q),
o2v ov
— -AveL2(Q), v(x,T)=0, —(x,T)=0.
dt2 dt
The problem fits the conditions of Section 9.4. Therefore we deduce
from (9.48) that
d2u
(9.49) —--Au = f
dt2
in the sense of distributions in Q,
(9.50) u(x,0) »««>(*)wl)),
(9.51) i^.(*,o)-«,(*)««).
01
Furthermore, we have the boundary condition
u = 0 on Et
((1)) According to (9.8a), u(.,t) -> uQ in [H, //_1 (Q)]1/2, space which coincides
with (Hyj-(Q))' when r is regular (see Chapter 1, Theorem 11.7 and Remark 12.1).
((2» According to (9.8a), u'(.,t) = — (.,t)-*u1 in [H-1 (Q), D (A)']1/2,
where ° *
D(A) = {v\v£H\(Q),Av£L2(Q)}.

294 9. Equations of the Second Order in t; Transposition
which is contained in (9.47), but in a formal manner: indeed, formally
integrating by parts in (9.47) and taking account of (9.49), (9.50), (9.51),
we obtain
(9.52)
u—-do = 0, i = rx]oj[.
dv
But this is formal on two accounts:
(i) no regularity hypothesis is made on 2\
(ii) if 2 is assumed regular, the integrations by parts would have to
be justified in a certain sense.
The justification of (ii) will be made in Chapter 5 of Volume 2; but
without regularity conditions onZ\ there exists (at the moment) no other
interpretation of the condition t(u = 0 on 2" than equation (9.47) itself.
9.6.2 Example 2
Let ii be defined as in the preceding example. We take H = L2 (ii)
again, and
V = H20(ii)
and then
V = H~2(ii).
We take
(9.53)
a(t;u,v) = AuAvdx.
The general theory applies, for, if A > 0,
\Av\2 + X\v\2*«\\v\\2H2m, <%>0,
So let /, u0, ul be given with
feL2(0,T;H-2{Q)) i.e.
VveH20(Q).
(9.54)
,_,.+ £ «.+i.','»
1 = 1 dxt ij dxtdxj
fo.ft, fueL2(Q),
u0 e L2 (ii),
uxeH'l(Q).
Then there exists a unique function u e L2 (Q) such that
(9.55)
(9.56) ul—j + Z*v\dxdt= [j(t),v(t)]dt +
+ [«!,»«»]-
dv
«o.-77«>)
01

9.6 Examples 295
for every function v el2(0, T; Hq(Q)) such that
dv d2v dv
— €L2(Q), — + A2veL2(Q), «(*,7) = 0. — (*. T) = 0,
ot dt2 dt
where [, ] denotes the antiduality between H~2(Q) and Hq(Q) (and
the scalar product in L2(Q)).
Then u satisfies
d2u
(9.57) —r + A2u=f)
(9.58) u(x,0) =u0(x)«"\
and
(9.59) 4t^'°)=miW((2)'-
ot
Remark 9.16. The present example shows that the hyperbolic character
of the wave operator does not intervene in Example 1. □
Remark 9.17. We have not been able to give the case in which u is
non-zero on E, in the preceding examples. This will be done, when E
is regular, in Chapter 5, Volume 2 (see, in particular, Section 1.2 of
Chapter 5). D
9.6.3 Example 3
Again let Q be an arbitrary bounded open set in Rn and take
H = L2{Q)t V = Hl{Q).
Then V is not a space of distributions on Q. In order to obtain the
"usual" interpretations, we may use the space Sl(Q) as defined in
Section 4.7.2.
We take a{t)u,v) as in (9.44).
Let /, u0, % be given with
(9.60) feL2(0,T;3-l(Q)),
(9.61) u0eL2(Q), u^S-^Q);
then, since V <= 51 (Q), we may take
_ T
dv
(9.62) (L,v)= f </(*), »(<)> dt + <«lf »(0)> -
0
«o.-r- (0)
ot
«*» With u (t) -+ u0 in H'1 (Q) if J1 is sufficiently regular and in [H, H"2 (Q)]lf2
in any case.
((2)) With u'q _+ Ui in [H-2(Q),D(AY]1/2, where
D(A) = {v\veH2Q(Q),A2veL2{Q)}.

296 9. Equations of the Second Order in t\ Transposition
in (9.5), where the brackets denote the duality between S"X(Q) and
S1 (Q) and [, ] denotes the scalar product in L2 (Q).
From which we deduce the existence and uniqueness of u e L2 (Q)
satisfying
(9.63) u (—\ - Jv\ dx dt = (L, v)
H
for all ^eI2(0J;F (Q)) such that
dv d2v dv
— eL2(Q), —-Av = <peL*(Q), v(x,T)-0, _(*,r)=0
Ol Ol 01
and
d2
2 [v{t),w] + a(v(t)tw) = [<p(t)tw] VweH^Q).
The function
(9.64)
(9.65)
Finally, in
(9.66)
u then satisfies
Au = f,
dt2 '
'
' u(x,0) =Mo(*)<(1)).
-^-(*,0) = M*)<(2))
01
a formal way
du
= 0 on E
dv
Remark 9.18. See also the example given in Remark 6.1, Chapter 5,
Volume 2. D
9.6.4 Example 4
Let Q be a bounded open set with regular boundary r, and assume
that ro is a subset of r.
Let rx be the complement of ro in r.
Take H = L2(Q) and
(9.67) v = {v | v e H1 (Q) tv = 0 on T0} ((3)).
((D) with u(t) -> u0 in [&(£)), H]'m.
((2)) with u^t)^Ul in [D{A),H1{D)],1,2, where
D(A) = {v\v€H1(Q),Av€L2(Q) and (—Av,w)=a{v,w), V^G//1^)}.
((3)) i.e. the restriction of y0 v to ro vanishes, y0 v being the trace on r
defined according to Chapter 1, Section 8.

9.6 Examples
297
Then (except if ro = r, in which case we have the Dirichlet
problem) V is not a space of distributions on Q.
Again let
" f du dv
a(t;u,v) = £ — — dx\
i = i J 3^| d%i
then the general theory applies.
For the interpretation of the corresponding problem, we meet the
same difficulty as in Section 9.6.3 above. We use the space Sro(Q) as
defined in Section 4.7.3.
In (9.5), we may now take (L, v) as given by (9.62), with this time
(9.68)
and
(9.69)
feL*(p9T;5f0l(Q))
u0 e L2 (Q), Ui e Srol (Q).
Then there exists a unique u e L2 (Q) such that
(9.70) \ul^-2v\dxdt = (L9v)
Vwel^OJ;//1^)), —eL2(Q), v = 0 on rox]0, T[,
8t
V -Av = <peL2{Q), v{x,T)=0, -1(*,T)=0,
dt2
dt
(9.71)
d2
—-[v(t),w] + a(v{t),w)=* [<p(t),w], VweH'iQ),
at
w = 0 on ro.
The function u satisfies
d2u
dt2
- Au = / in Q,
(9.67), and the mixed conditions
u = 0 on rox]0,T[t
(9.72)
du
dv
= 0 on TiXjO,^;
these conditions in a "weak" sense (tied to (9.70)). D

298 9. Equations of the Second Order in /; Transposition
9.6.5 Example 5
Let Q be a bounded open set in R2. Take (as in Example 4.7.6)
H = L2(Q),
v\veL2(Q), -^-eL2(Q), ^-eL2{Q) \,
(9.74)
a(t] u,v) =
dxl
du dv
dxx dxx
dx +
dx2
d2u d2v
dxl dxl
dx.
The general theory applies. For the interpretation of the problem,
one possibility is to use the space K(Q) introduced in Section 4.7.6.
We then take (L,v) defined by (9.62) with
(9.75) feL2(ptT'fK>{Q))t
(9.76) u0eL2(Q), u.eK^Q).
The corresponding solution u e L2 (Q) satisfies
( d2u d2u d*u
^r + T-*=f in Q>
(9.77)
dt2 dx2 ' dx\
u(x, 0) = u0>
du
dt
(x,0) = ult
and, in a weak sense, boundary conditions which are, when Q is the
rectangle D = ]0, ar[ x ]0, a2[:
u = 0, on the sides of r parallel to the %2-axis,
du
u =
dx2
= 0, on the sides of T parallel to the ^-axis. D
Remark 9.19. In this example, the number of derivatives in V
depends upon the direction of differentiation.
Other situations may be encountered; for example, let
v\veL2(Q),nv=-^2- VYeL2(D)
and
dxl dx2
a(t]u,v)= uuUvdx.
Then the corresponding partial derivative operator will be
+ D2.

10.2 Existence and Uniqueness Theorem
299
In this case, the choice of a space K(Q) (different from L2(Q)\)
does not seem immediate (see Problem 13.12). D
10. Schroedinger Type Equations
10.1 Notation
Again the operators A (t) are given by the family a(t; u,v) with the
properties (8.1), (8.2). In this section we shall consider the (Schroedinger
type) equations
(10.1) iA(t)u + u' = f
with the initial condition
(10.2) u(0) = u0.
10.2 Existence and Uniqueness Theorem
Theorem 10.1. Hypotheses (8.1), (8.2) are assumed to be satisfied.
Let f be given with
df
(10.3) / e L2(0, T) H), — = /' e L2(0, T\ V)
dt
and let u0 be given with
(10.4) u0 e V.
There exists a unique u satisfying
(10.5) ueL2{0,T;V)
and (10.1), (10.2).
Remark 10.1. If (10.5) holds, then, according to (10.1),
(10.6) u' =/- iA(t)ueL2(0,T; V),
so that (10.2) has meaning. D
Proof of Theorem 10.1. As for the proof of Theorem 8.1, we first
give the a priori estimates which imply (with the standard procedures)
the existence of a solution.
1) A priori estimates. Multiplying (10.1) by u(t), we obtain
i [A (t) u(t),u (t)] + \u' (t), u (t)] = [/ (t), u (*)],
from which, taking twice the real part of the two members:
(10.7) — \u(t)\2 = 2Re[f(t),u(t)]

300 10. Schroedinger Type Equations
whence t
(10.8) \u(t)\2 = \u0\2 + 2Re J [/{a), u{a)] da
o
and therefore
t t
(10.9) \u(t)\2 = \u0\2 + j \f(a)\2do + J \u(a)\2da.
0 0
This, together with Gronwall's Lemma, implies
(10.10) \u(t)\2 ^ cl\u0\2 + j \f(a)\2da\, O^t^T.
Now, multiply "formally'' (10.1) by u', taking twice the imaginary
part of the result; we obtain
a(t;u(t), u'(t)) + a(t;u'(t),u(t)) = 2 Im [/(*), «'(*)],
from which we get
d
(10.11) —a(t]u(t),u(t)) - a'(t;u{t),u{t)) = 2 Im [/(*), u'(t)].
dt
Integrate (10.11):
t
a(t]u(t),u(t))= a(0;uOfuo) + J a'{a\u{a),u{a))da +
o
t
+ 2Im j[f (a), u'(a)] da.
(10.12)
But
t
f [/ W, «' (*)] ^cr = [/ (<), «(*)] - [/ (0), u0] - J [/' (a), «(a)] da
(note that /(0) e [H, V']1/2 <= F). Therefore
J [/(*).«»] iff
^ II/WIIf'II«WII + ll/(0)||F.||«oll +
t
j\\f'(a)\\v.\\u(a)\\da
+
and (10.12) yields:
*\\u(t)\\2 ^ C t\\u0\\2 + j \\u(a)\\2 da + j \\f'(a)\\2v, da)
+ 2\\f(t)\\vr\\u(t)\

10.2 Existence and Uniqueness Theorem 301
The last term is bounded above by
t
^\W)\\2 + -\\f(t)\ti- £^\\u(t)\\2 + -C [(\f(a)\2 + \\f'(o)\\l.)do
2 (x 2 oc J
o
and therefore
(10.13) , , t v
ll«WH2 ^ c/|KII2 + Jll«WII2^ + /(l/WI2 + \\i'{<y)\\2v)do\.
This inequality, together with Gronwall's Lemma, implies:
(10.14) , T .
\\u(t)\\*zcl\\u0\\2 + j(\f(o)\2 + \\n<y)\\2v.)da\, O^t^T. D
2) Existence of a solution. We may use the preceding estimates
together with the method of Faedo-Galerkin (for example) to show
existence (we could also use parabolic regularization, in analogy with
Section 8, as will be done explicitely in Chapter 5, Section 12, Volume 2).
So let wx, . . ., wmi . . . be a "basis" of V (see Section 8; V is assumed
to be separable) and let
m
i = l
be given by
(10.15) (u'm(t), wj) + i a(t;-um(t),wj) = (f(t),Wj), lgj'^m,
m
(10.16) gim(0)=|im, El^w^Mo in V.
1=1
Then the same bounds as in 1) (see (10.14)) show that
(10.17) um remains in a bounded set of L°°(0, T\ V), as m -* oo.
We extract u^ -► u weak star in L°°(0, T; F) and verify that u is
a solution of
du
(10.18) ii4 (/)«+-— = /, «(0)=«0.
at
Then it follows from (10.18) that
dii
— eLx(0,T;V).
at
According to the uniqueness (which is shown separately), we have
u = u and therefore existence. □

302 11. Schroedinger Type Equations; Transposition
Remark 10.2. Thus we have shown more than Theorem 10.1, that
is, the existence of a solution satisfying
(10.19) WGL°°(0,r;F),
(10.20) ^eL^OJiF).
In fact, by the method of Section 8.4, we can show that u is a
continuous function of [0, T] -* V and u' a continuous function of [0, T] -►
-> V. U
Remark 10.13. We shall see other regularity results — established
by a somewhat different method — in Chapter 5, Volume 2, and then
Mfc-regularity results in Volume 3. D
Now we shall briefly investigate the "transposition " of Theorem 10.1.
11. Schroedinger Type Equations; Transposition
11.1 Adjoint Isomorphism
We consider the adjoint problem:
(11.1) -\A{t)v - v' = (p
(11.2) v(T) =0,
where <p describes the space 0 defined by
(11.3) 0 = {<p \ <p eL2 {0, T; H), <p' eL2(0, T; V), <p(0) =^(T)=0}.
Remark 11.1. In (11.3), we have imposed the conditions "(p(0)
= (p(T) = 0" so that the space @{]0, T[\ H) be dense in 0, 0 being
provided with the natural (hilbertian) norm:
/ T \ 1/2
M(M')I2 + llv'WIlJO^J d
We define:
IX = space described by the solution v of (11.1), (11.2) as q>
describes 0.
Providing X with the topology carried over by the topology of 0
in the mapping <p -► v, we have (done what was necessary to have)
d
(11.5) —\A is an isomorphism of X onto 0. D
dt

11.3 Choice of L
303
11.2 Transposition of (11.5)
Theorem 11.1. Assume (8.1), (8.2) to be satisfied. Let L be a continuous
antilinear form on X. There exists a unique u e &' with
(11.6) (u, -iA(t)v -v') = (L,v), V^el,
(where the first parenthesis denotes the duality between &' and 0
and the second between X' and X). D
From Remark 11.1 and the Hahn-Banach Theorem it follows that
(11.7)
dul
every u e &' may be written, non-uniquely, as u = u0 -\ ,
dt
w06l2(OJ;i/), ^el^OJjT). D
We still have to choose L.
11.3 Choice of L
Analogously to Sections 4.5. and 9.3, we define the space 3 by
(q being the function defined by (4.23)):
(11.8) 3={v\v6L2(0,T',V),qv'6L2(0,T)V'),v(T)=0}
I provided with the norm
1/2
j(\\v(t)\\2+\\QV'(t)\\2V,)dt\ ,
which makes it a Hilbert spaceJ.
We see again that ^(]0, T[; V) is dense in 3 and therefore 3' = anti-
dual of 3, is a space of distributions on ]0, T[ taking their values in V;
furthermore, every f eS' may be written, non-uniquely, as
(11.9) / = /0+ir(^/l)> With /0eL2(0,T;F), A eL2(0, T; F).
at
Finally, we may take
(11.10) L(v) = (f,v)s.9S+(uo,v(0)), with feS' and u0eV\
in (11.6), ( , )s\s denoting the scalar product between 3' and 3'. Then,
formally, (11.6) yields:
u e 0' (the structure of which is given by (11.7))
(11.11) ] \A{t)u + u' =/, given by (11.9)
u(0) = u0, u0e V. D

304
12. Comments
Remark 11.2. It is possible [at least under hypothesis (9.14)] to give
an interpretation of the type given in Section 9 to (11.11). But we do
not specify the results here; see also Problem 13.12. D
Remark 11.3. Of course, we can interpolate, as, for example, in
Remark 9.15. D
Remark 11.4. Examples analogous to those of Sections 4.7 and 9.6
could be given here. □
12. Comments
The introduction of weak solutions of evolution equations is
classical, in particular since the works of Leray [2—4] (in connection with
nonlinear Navier-Stokes equations) and Sobolev [3]: see Ladyzenskaya
[1—3], Ladyzenskaja-Vishik [1], Lions [13], Treves [1, 2], Vishik [4].
The presentation of Sections 1 — 3 follows the paper of Lions [27]
(which contains the case of variational inequality problems in an
analogous framework; for more general results in this direction, see Brezis
[1. 2]).
For the examples of Section 4, see also Browder [4, 10, 11], and
Lions [13]. All the examples given here correspond to the case for which
D(A{t)) (suitably defined, see Kato [1—3]) is independent of t\ when
D(A(t)) depends on t, and in a Hilbert setting, see Baiocchi [3, 4],
Bardos [1], Brezis [3], CarroU [1, 2], G. Cooper (1], Lions [13, 18, 22, 23].
We have not touched upon the study of corresponding non-homogeneous
problems, see Problems 13.6, 13.8.
In the Banach space setting and through the use of semi-group
theory (see Hille-Phillips [1], Yosida [2]), first-order operator equations
have been studied by Yosida [1], Kato [1, 2, 5], Kato-Tanabe [1],
Poulsen [1], Sobolevski [1—7], Tanabe [1] and S. Krein [4].
Results of the type given in Section 5, have been obtained by
Kaplan [1] via different methods.
The method of elliptic regularization used in Section 7 was
introduced in this form by Lions [21]. For the case of elliptic-parabolic
equations, an analogous regularization method was introduced by
Oleinik [4]. Another method consists in replacing M + A by M +
H — = Mh\ then one applies elliptic variational theory (Chap-
h
ter 2, Section 9) and lets h go to zero.
Of course, there are many possible choices for the space in which we
seek a solution, not only in the "space variables" (abstract or not) but
also in the time variable.
Thus Sobolevski has studied the "abstract Cauchy problem" in
spaces satisfying weighted Holder conditions in t (see Sobolevski [3])

12. Comments
305
and in Lp or Lorentz spaces (still in /) (see Sobolevski [4, 5]). Along these
lines, see also Da Prato-Giusti [1, 2].
Equations of the second order in t are studied in Sections 8 and 9.
Regularity results of the same type as those in Theorem 8.2 are given
by Baoicchi [7], Lions [21], Torelli [1], Strauss [1]; we have followed
the presentation of Strauss for Lemmas 8.1, 8.2 and 8.3.
Concerning the existence and uniqueness theorems for the Cauchy
problem, we also call attention to the method of Baoicchi, who has
obtained the results directly, without duality, under slightly different
hypotheses; this direct method is valid for first and second order
equations (see Baiocchi [6, 7]) and Schroedinger type equations (see
Pozzi [1]).
Other existence and uniqueness results, using esentially different
methods (Dunford integrals and interpolation) are gives in Grisvard
[6, 7, 9] and (still by different methods) in Da Prato [4, 5].
Operator equations of the second order in t have been studied by
semi-group methods in Pogorelenko-Sobolevski [1], and Raskin-Sobo-
levski [1]; also, see the work of S. Krein [4].
The choice of the particular scalar product (u, v)# (given in (8.36))
is the analog, when the coefficients depend on t, to what is done in the
case of independence of t, to show (according to Yosida [1]) that
is the infinitesimal generator of a group (see also Lions [7]). It is
important to choose a suitable scalar product on Jf; for example, the property
for
—Co)
to be the infinitesimal generator of a semi-group in Jf, is independent
of the scalar product] but in order to apply the theorem of Hille-Yosida,
it is much easier to come back (if possible) to the case where the
semigroup is a contraction semi-group — and this is precisely the object of
the "suitable" choice of the scalar product on Jf; see also
J.A.Goldstein [1], Mizohata [1].
For the case of operator equations with lag, we refer to Artola [1],
Baiocchi [7] and Pozzi [1].
The case of stochastic operator equations would be treated by the
same type of methods; in fact, this is true for all the results of this
book, but the rather long preparation of the framework would have
made the presentation too heavy.
Still in the way of "general evolution equations", we point out the
following studies (the subjects of which are not touched upon here):
U -1
A 0

306
12. Comments
1) the theory of semi-group distributions: see Da Prato-Mosco [1,2],
Foias [1], Fujiwara [1], Lions [12], Peetre [9], Shiraishi-Hiraka [1],
Yoshinaga [1, 2] and Chazarain [1, 2];
dk
2) the Cauchy problem is not well-posed for the operators A + —r
when k > 2, except for the (uninteresting) case where A is a bounded
* dJ
operator: see Fattorini [1], Chazarain [1]; for the operators ^Aj—r,
j-Q dtJ
unbounded Ajt see very partial results in the following chapters.
But for problems different from the Cauchy problem, the general
d2m+1
result of Section 1 applies; for example, for the operator A (t) -\
fi2m+l dt
on a finite interval (0, T), by taking A = 2m+l , the domain of A
being defined by ua)(0) = 0, uU)(T) = 0, Og/gw-1. This
reproduces (using the intermediate derivatives theorem) the results of
Kadlec [1], Cattabriga [5], Grisvard [7], who also investigates the non-
Hilbert cases;
3) for the general study of the solutions of the equations
du
Au + = 0
dt
(behaviour at infinity, asymptotic representation of the solutions,
norm convexity properties of solutions . . .) according to the spectral
properties of the unbounded operator A, see Agmon-Nirenberg [1, 2],
Geymonat [5], Lax [1], Lioubich [1], Zaidman [1, 2];
4) for the study of almost periodic solutions of evolution equations,
see Amerio-Prouse [1] and the bibliography of this work;
5) for the "operator" generalization of Sturm-Liouville problems,
see Kostiouchenko-Levitan [1];
6) for the use of Wiener type integrals for evolution equations, see
Daletski [1], Nelson [1] (in these works, a result of Trotter [1] is used,
a result which incidently can also be used as a starting point for numerical
approximation methods; see generalizations in Bardos [1]);
7) certain properties — in particular of uniqueness — remain valid
du
when the equality + A u = / is replaced by an inequality satisfied
dt ii A„
au
(in a suitable norm) by -r- + Au — f
nabe [3]; "
see Friedman [3], Ta-

13. Problems
307
8) the problems mentioned in 7) should not be confused with the
variational evolution inequalities, such as are studied in Lions-Stampac-
chia [1], Brezis [1—3], Brezis-Lions [1];
9) we call attention to the study of operator equations in locally
convex topological vector spaces; see Yosida [3];
10) for transport equations, see Jorgens [1] and S. Krein [4];
11) we have left aside the question of (eventual) analyticity of
solutions in t; we shall come back to this point in Volume 3; consult
Kato [3], S. Krein [4, 5], Yosida [2];
12) for "scattering" theory, see Kato [5], Lax-Phillips [2];
13) we also call attention to the study, in Baouendi-Grisvard [1],
of the operator
du d2mu
dt v dx2m
the type of which changes according to the sign of x\ this study has
been carried on to analogous nonlinear cases by Bardos-Brezis [1].
Also consult the Comments to Chapters 4 and 5 (Volume 2).
13. Problems
13.1 In Theorem 1.1, the semi-group G(s) operates in 34?, in i^
and in "K'. What can be said when G(s) is a contraction semi-group in
3tf, but does not operate in V and ^'? (see also 13) in Section 12).
13.2 The study of non-homogeneous problems for operators
where A (t) is a singular integro-differential operator, may be of interest.
For example for the case (which comes up in applications) where
A (t) = A is given by
i
A = p.v. <p'(y)dy,
J *-y
-i
see Cherruault [1] and the bibliography of this work.
13.3 Extension of the results of Sections 4 and 5 to operators
d
A (t) + — in Banach spaces by the use of semi-group theory; Yosida [2],
dt
Kato [1, 2], Kato-Tanabe [1], Sobolevski [1-3], Tanabe [1].

308
13. Problems
13.4 In the setting of Section 4, when A (t) verifies
Re(4(*KtO + A(*)M2^*||^
with XeL}{0tT) (see Lions-Raviart [1], Baiocchi [5]); see also
Problem 17.15 in Chapter 4 of Volume 2.
13.5 Case for which the space V of Section 4 is replaced by a
measurable family of Hilbert spaces V (t); see the bibliography given in
the Comments to this chapter.
d2
13.6 Problem analogous to 13.5, but for the operators A (t) -f .
13.7 Problems 13.5 and 13.6 contain, as particular cases on the
examples, problems in non-cylindrical open sets; then see the problems
of Chapters 4 and 5, Volume 2.
13.8 Obtainment of the results of Sections 4, 5 and 6 by passage
to the limit e -* 0 starting from non-homogeneous "elliptic" problems,
by the application of elliptic regularization (Section 7).
13.9 Question analogous to 13.8 for the results of Sections 9
and 11; for "elliptic regularization" corresponding to Section 8, see
Strauss [1].
13.10 It would be of interest to systematically study the " smallest
possible" spaces K(Q), such that V cz K(Q), SJ{Q) is dense in K(Q)f
when V is defined by
V = {v | v e L2 (Q), Atv e L2 (Q), and boundary conditions},
where the At's are differential operators with constant or variable
coefficients. The spaces 3s (Q), studied in Chapter 2 and the examples
of Section 9.6 are particular and partial cases. We ignore whether there
exists a minimal space with these properties.
13.11 General interpretation of equation (9.5) in case feS' (see
also Section 11, Chapter 5, Volume 2).
13.12 Problem analogous to 13.11 for the case of Schroedinger
equations (Section 11).

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Additional Bibliography
Chapter 1.
For the general theory of interpolation of linear operators one can also consult
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Gagliardo, E.
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One can extend the theory (or, at least, part of it) to non linear operators
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For the case when the non linear operator is Lipschitzian in the two couples
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Applications and extensions arc given in
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New interpolation results for non-Banach spaces are given by
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For the interpolation between subspaces (cf. Problem 18.5) cf.
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(1969).
Chapter 2.
For an approach to boundary value problems of elliptic type using pseudo-
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Bibliography (cf. particularly Vhishik-Eskin [1], [4], [5]), to
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Mat., Symposia Mat., Roma, 1971.

350
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Kr6e, P.
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For papers which are directly related to the content of Chapter 2, let us refer
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Torelli, A.
1. Sulla teoria variazionale dei problemi ai limiti ellittici. Rend. 1st. Lombardo
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These papers give solutions to the general question of Problem 11.5. For
variational methods cf. also
Kr6e, P.
3. Application des m^thodes variationnelles aux Equations de convolution.
C. R. Acad. Sci. Paris, A, 268, 1193-1196 (1969).
For Green's formula and adjoint problems cf.
Troisi, M.
1. Sulla nozione di problema aggiunto per i problemi al contorno relativi ad
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and 164-215 (1968).

Additional Bibliography
351
For questions related to Sections 6 and 7 (trace theorems and
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Baouendi, M. S., Geymonat, G.
1. R6sultats de dualite* dans les problemes aux limites lin^aires elliptiques (I),
J. of Differ. Equat. (to appear).
This paper also considers a problem similar to the "best" choice of the
space Kr(Q) (cf. Section 6.3).
Boutet de Monvel, L., Geymonat, G.
1. Already quoted in this Additional Bibliography.
Goulaouic, C, Grisvard, P.
1. Existence de traces pour les 616ments d'espaces de distributions d6finis
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Kr£e, P.
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Martsinkovsa, G.
1. Linear functionals over Sobolev spaces and boundary problems generated
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Roitberg, Yu. A.
1. On boundary values of generalized solutions of elliptic equations. Dokl.
Akad. Nauk 188, 41-44 (1969) [Soviet Math. 10, 1079-1083 (1969)],
2. Green's formula and the homeomorphism Theorem for general elliptic
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Rushchitskaya, S. O.
1. A theorem on homeomorphisms for an elliptic differential operator and
pseudodifferential boundary conditions. Ukrainskii Math. J. 21, 268 — 273
(1969).
For "fractional powers" of operators, cf.
Fujiwara, D.
2. Lp-theory for characterizing the domain of the fractional powers of —A in
the half space. J. Fac. Sc. Univ. Tokyo 1, 15, 169—177 (1968).
3. On the asymptotic behavior of the Green operators for elliptic boundary
problems and the pure imaginary powers of some second order operators.
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Komatsu, H.
1. Fractional powers of operators, III. Negative powers. J. Math. Soc. Japan
21, 205-220 (1969).
2. id. IV, Potential operators. J. Math. Soc. Japan 21, 221—228 (1969).
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352 Additional Bibliography
Shimakura, N.
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derives obliques. J. Math. Kyoto Univ. 9, 363-379 (1969).
Yoshikawa, A.
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de plusieurs espaces de Banach. J. Fac. Sic. Univ. Tokyo, Sect. 1, 16, 407
to 468 (1970).
2. An abstract formulation of Sobolev type imbedding theorems and its
applications to elliptic boundary value problems. J. Fac. Sic. Univ. Tokyo,
Sect. 1, 17, 543-558 (1970).
3. An operator theoretical remark on the Hardy-Littlewood-Sobolev inequality.
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4. Fractional powers of operators, interpolation theory and imbedding
theorems, to appear.
For the regularity of the solution of elliptic problems at "corners" using an
"operational" method, cf.
Grisvard, P.
1. Equations op^rationnelles et problemos aux limites dans les domaines non
r^guliors. Proc. Int. Congress of Math., Nice, 1970.
For Dirichlet's problem in the half-space, cf.
Brezzi, F.
1. Sul problema di Dirichlet ncl semispazio. Ann. Mat. Pura Appl. 4, 86, 261
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For the problem of "transmission" cf.
Kree, P.
4. Problemes de transmission pour des operateurs pseudo-diff^rentiels ellip-
tiques. C. R. Acad. So. Paris, A, 269, 699-701 (1969).
For the problem of the "oblique" derivative and generalizations cf.
Bitsadze A. V.
1. Linear non-Fredholm elliptic boundary value problems. Proc. Int. Congress
of Math., Nice, 1970.
Egorov Ju. V., Kondratiev, V. A.
1. On a problem of oblique derivative. Mat. Sbornik 78, 148 — 176 (1969).
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1. Problemi di derivata obliqua per equazioni ellittiche in due variabili. Boll.
U.M.I., 3, 22, 505-526 (1967).
For elliptic operators which degenerate at the boundary, cf.
Baouendi, M. S., Goulaouic, C.
1. R^gularite et th^orie spectrale pour une classe d'operateurs elliptiques
d6g£ne>6es. Arch. Rat. Mech. Anal. 34, 361-379 (1969).
2. Etude de I'analyticite et de la regularite Gevrey pour une classe d'op^rateurs
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Additional Bibliography 353
3. Regularite analytique et iteres d'operateurs elliptiques degeneres:
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Bolley, P., Camus, J.
1. Etude de la regularite de certains problemes elliptiques degeneres dans des
ouverts non reguliers, par le methode de reflexion. C. R. Acad. Sc. Paris,
A, 268, 1462-1464 (1969).
2. Sur certains problemes aus limites, elliptiques et degeneres. C. R. Acad.
Sc. Paris 271, 980-983 (1970).
Hanouzet, B.
1. Regularite pour une classe d'operateurs elliptiques degeneres du deuxieme
ordre. Le Mathematiche 24, 450—491 (1969).
Shimakura, N.
2. Problemes aux limites generaux du type elliptique degenere. J. Math. Kyoto
Univ. 9, 275-335 (1969).
Troisi, M.
1. Problemi ellittici con dati singolari. Ann. Mat. Pura Appl. 4, 83 (1969).
2. Ulteriori contributi alio studio dei problemi ellittici con dati singolari.
Ricerche Mat. 19, 9-25 (1970).
Vishik, M. I., Grusin, V. V.
1. On a class of degenerate elliptic equations of higher order. Mat. Sbornik
79, 3-36 (1969).
2. Boundary value problems for elliptic equations that degenerate at the
boundary. Mat. Sbornik 80, 455-491 (1969).
3. Degenerate elliptic differential and pseudo differential operators. Uspehi
Mat. Nauk 154, 29-56 (1970).
For the asymptotic distributions of eigenvalues, cf.
Agmon, S.
1. Asymptotic formula with remainder estimates for eigenvalues of elliptic
systems. Arch. Rat. Mech. Anal. 28, 165—183 (1968).
Hormander, L.
1. The spectral functions of an elliptic operator. Acta Math. 121, 193—218
(1968).
For elliptic systems, cf.
Roitberg, Ya. A., Sheftel', Z. G.
1. A theorem on homeomorphisms for elliptic systems. Mat. Sbornik 78, 446
to 472 (1969).
For "variational" second order elliptic equations with discontinuous
coefficients, cf.
Chicco, M.
1. Principio di massimo generalizzato e valutazione del primo autovalore per
problemi ellittici del secondo ordine di tipo variazionale. Ann. Mat. Pura
Appl. 4, 87, 1-10 (1970).

354 Additional Bibliography
Herv6, R. M. and M.
1. Les fonctions surharmoniques assoctees a un operateur elliptique du second
ordre a coefficients discontinus. Ann. Inst. Fourier 19, 305 — 359 (1969).
Marino, A., Spagnolo, S.
1. Un tipo di approssimazione dell'operatore J ^/(fl/jW)^j con operatori
n //./'
2Dj(P(x)Dj). Ann. Sc. Norm. Sup. Pisa 23, 657-673 (1969).
/./
Spagnolo, S.
1. Una caratterizzazione degli operatori differenziali autoaggiunti del 2° ordine
a coefficienti misurabili e limitati. Rend. Sem. Mat. Padova 39, 56—64
(1967).
2. Sulla convergenza di soluzioni di equazioni paraboliche ed ellittiche. Ann.
Sc. Norm. Sup. Pisa 32, 571-597 (1968).
For " non-variational" elliptic equations, cf.
Campanato, S.
1. Equazioni ellittiche non variazionali a coefficienti continui. Ann. Mat.
Pura Appl., 1971.
2. Un risultato relativo ad equazioni ellittiche del secondo ordine di tipo non
variazionale. Ann. Sc. Norm. Sup. Pisa 21, 701—707 (1967).
Chicco, M.
2. Equazioni ellittiche del secondo ordine di tipo Cordes con termini di ordine
inferiore. Ann. Mat. Pura Appl. 4, 85, 347—356 (1970).
For the numerical solution of elliptic problems and error estimates, we refer to
Aubin, J. P.
1. Approximation of non homogeneous elliptic boundary value problems,
New York: Wiley 1971.
Bramble, J. H., Schatz, A. H.
1. Least square methods for 2mih order elliptic boundary value problems
(to appear).
2. Rayleigh-Ritz-Galerkin methods for Dirichlet's problem using subspaces
without boundary conditions. Comm. pure appl. Math. 23, 653—675 (1970).
Ciarlet, Ph., Wagschal, C.
1. (to appear).
Fix, D., Strang, G.
1. Book on Finite elements (to appear).
Guglielmo, F. di
1. Construction d'approximation des espaces de Sobolev sur des reseaux en
simplexe. Calcolo 6, 279—331 (1969).
2. M^thodes des elements finis: une famille d'approximation des espaces dc
Sobolev par les translates dc p fonctions. Calcolo 7, 185 — 234 (1970).
Zlamal, M.
1. On the finite method, Numer, Math. 12, 394-409 (1968).

Additional Bibliography 355
2. On some finite element procedures for solving second order boundary value
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These papers give solutions to Problem 11.11.
Chapter 3.
For the general approach to abstract differential equations, cf.
Baiocchi, C.
1. Teoremi di regolarita per le soluzioni di equazioni differenziali astratte.
1st. Naz. Alta Mat., Symposia Math., Roma, 1970.
Carrol, R.
1. Abstract Methods in Partial Differential Equations, New York: Harper
& Row 1969.
Da-Prato, G.
1. Weak solutions for linear abstract differential equations in Banach spaces.
Advances in Math. 5, 181-245 (1970).
Kr6e, P.
5. Synthase et extension de certaines theories variationnelles lin6aires. C. R.
Acad. Sc. Paris, A, 271, 457-460 (1970).
For problems with 1st order /-derivative and variable spaces V (t), cf.
Baiocchi, C.
2. Problemi misti per l'equazione del calore. Rend. Sem. Mat. Fis. Milano,
1970.
Carroll, R. W., Cooper, J. M.
1. Remarks on some variable domain problems in abstract evolution equations.
Math. Ann. 188, 143-164 (1970).
Carrol, R., Mazumdar, T.
1. Solutions of some possibly noncoercive evolution problems with regular
data (to appear).
For questions related to Problem 13.4 and its applications cf.
Baiocchi, C.
3. Soluzioni deboli dei problemi ai limiti per le equazioni paraboliche del
tipo del calore. Rend. 1st. Lombardo Sc. Let., A, 103, 704—726 (1969).
Marino, M.
1. Sull'esistenza delle soluzioni deboli dei problemi al contorno per operatori
parabolici. Le Matematiche 13, 387—407 (1968).
For differential equations with "delay" terms
Artola, M.
1. Sur les perturbations des Equations devolution. Application a des problemes
de retard. Ann. Sc Ec Norm. Sup. 4, 2, 137-253 (1969).

356 Additional Bibliography
Comincioli, V.
1. Problemi periodici relativi a equazioni d'evoluzione paraboliche con termini
di ritardo . . . Rend. 1st. Lombardo Sc. Let. A, 104, 356—381 (1970).
2. Ulteriori osservazioni sulle soluzioni del problema periodico per equazioni
paraboliche lineari con termini di perturbazionc. Rend. 1st. Lombardo Sc.
Let., A, 104, 726-735 (1970).
A general regularity theorem has been given in
Bardos, C.
1. A regularity theorem for Parabolic equations. J. Functional Analysis
7, 311-322 (1971).
For problems with second-order ^-derivative cf.
Kato, T.
1. Linear evolution equations of "hyperbolic" type. J. Fac. Sci. Univ. Tokyo
17 (1, 2), 241-258 (1970).
and the papers:
Carroll, R., State, E.
1. Existence theorems for some weak abstract variable domain hyperbolic
problems (to appear).
Fattorini, H. O.
1. Uniformly bounded cosine functions in Hilbert space. Indiana Univ. Math. J.
20, 411-425 (1970).
Giusti, E.
1. Funzioni coseno periodiche. Boll. U.M.I. 22, 478—485 (1967).
Goldstein, J. A.
1. Abstract evolution equations. Trans. Amcr. Math. Soc. 141, 159 — 185
(1969).
For various aspects of stochastic evolution equations we refer to
Bensoussan, A.
1. Filtrage optimal des systemes lin^aires, Paris: Dnnod 1971.
Bensoussan, A., Teman, R.
1. (to appear).
For the asymptotic behavior of the solutions of abstract differential equations
cf.:
Artola, M.
2. D6riv6es intermediates dans les espaces dc Hilbert ponders. Application
au comportement a Too des solutions des Equations devolution. Rend.
Sem. Mat. Padova 43, 177-202 (1970).
Pazy, A.
1. Asymptotic expansions of solutions of ordinary differential equations in
Hilbert space. Arch. Rat. Mech. Anal. 24, 113-218 (1967).

Additional Bibliography
357
2. Asymptotic behavior of the solution of an abstract equation and some
applications. J. Diff. Equat. 4, 493-509 (1968).
For somewhat related problems for variational inequalities we refer to the work
of H. Brezis mentioned in the bibliography and to
Lions, J. L.
4. Quelques m^thodes de resolution des problemes aux limites non lineaires.
Paris: Dunod, Gauthier-Villars 1969.
For applications cf.
Duvaut, G., Lions, J. L.
1. Sur les inequations en Mecanique et en Physique. Paris: Dunod 1971.
721/52/71 - 111/18/203

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